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PEH:Resistivity and SP Logging
Petroleum Engineering Handbook
Larry W. Lake, Editor-in-Chief
Volume V – Reservoir Engineering and Petrophysics
Edward D. Holstein, Editor
Copyright 2007, Society of Petroleum Engineers
Chapter 3B – Resistivity and SP Logging
- 1 Fundamentals
- 2 The Logging Environment
- 3 Electrode Resistivity Devices
- 4 Induction Logging
- 5 LWD Induction Tools
- 6 Induction vs. Laterolog Measurements
- 7 Microresistivity Logs
- 8 Formation-Resistivity Determination
- 9 Traditional Rt Methods
- 10 SP Log
- 11 Uses and Interpretation of Well Logs
- 12 Shaly Formations
- 13 Dual Water Models
- 14 Summary
- 15 Nomenclature
- 16 References
- 17 SI Metric Conversion Factor
Resistivity logging is an important branch of well logging. Essentially, it is the recording, in uncased (or, recently, even cased) sections of a borehole, of the resistivities (or their reciprocals, the conductivities) of the subsurface formations, generally along with the spontaneous potentials (SPs) generated in the borehole. This recording is of immediate value for geological correlation of the strata and detection and quantitative evaluation of possibly productive horizons. The information derived from the logs may be supplemented by cores (whole core or sidewall samples of the formations taken from the wall of the hole).
As will be explained later, several types of resistivity measuring systems are used that have been designed to obtain the greatest possible information under diverse conditions (e.g., induction devices, laterolog, microresistivity devices, and borehole-imaging devices). Many service companies offer resistivity-logging services, and most offer a Web-based catalog that describes each service.
Formation resistivity is a key parameter in determining hydrocarbon saturation. An electric current can pass through a formation because it contains water with enough dissolved ions to be conductive. With a few rare exceptions, such as metallic sulfides and graphite, dry rock matrix is a good electrical insulator. However, perfectly dry rocks seldom occur below ground level, so nearly all subsurface formations have finite, measurable resistivities because of the water in their pores, adsorbed onto their grain surfaces, or absorbed into a clay structure.
The resistivity of a formation depends on the resistivity of the formation water, the amount of water present, and the structure and geometry of the pores. The resistivity (specific resistance) of a substance is the electrical resistance measured between opposite faces of a unit cube of the substance at a specified temperature or, generally,
where R = resistivity in ohm•m, r = resistance in ohm, A = area in m2 , and L = length in m.
Conductivity, σ, is the reciprocal of resistivity, expressed in Siemens/m. To avoid the excessive use of small decimal numbers in well logging, conductivity is expressed in milliSiemens/m (mS/m), where 1000 mS/m = 1 Siemen/m, so
Formation resistivities are usually in the range of 0.2 to 1000 ohm•m. Resistivities higher than 1000 ohm•m are uncommon in most permeable formations but are observed in impervious, low-porosity formations such as evaporites. A few low-porosity hydrocarbon-bearing formations with almost no formation water can have resistivities as high as 20 000 ohm•m.
Formation resistivities are measured either by passing a known current through the formation and measuring the electrical potential (electrode or galvanic devices) or by inducing a current distribution in the formation and measuring its magnitude (induction devices).
Because resistivities cannot be read accurately over the entire measurement range when displayed on a linear scale, all resistivity logs are now presented on logarithmic grids, usually in 4 decades across two log tracks. This allows the display of readings from 0.2 to 2000 ohm•m, with a single curve covering the useful range of nearly all logs. A backup curve is used in the exceptional cases of readings outside that range. As resistivity data are used more and more digitally, and the log plots are mostly for reference, other formats are in wide use.
Most wireline resistivity-logging tools also have the ability to measure and record small differences in electrical potential that occur spontaneously in conductive muds as a continuous SP curve. The SP curve records the electrical potential differences generated by the interaction of formation water, conductive drilling fluid, and ion-selective shales. It has a wide range of formation-evaluation applications, including differentiating potentially porous and permeable reservoir rocks from impermeable clays and shales, defining bed boundaries, correlating nearby wells, indicating the shaliness of shaly formations, and estimating formation-water resistivity, Rw.
The Logging Environment
Measuring the resistivity of formations of interest is complicated by the invasion of drilling fluids into permeable rocks. The invaded fluid can displace some or all of the connate water or hydrocarbon present. The resistivity of the borehole is often much less than the formations of interest, sometimes by orders of magnitude. In addition, the formation often consists of rock layers with widely varying resistivity. Fig. 3B.1 shows these factors.
To measure the uninvaded portion of the rock from the borehole, a resistivity device must include a large volume of formation—there must be adequate sensitivity to the region of interest. Over the years, the main developments in resistivity-logging tools have been targeted at eliminating response from unwanted parts of the formation and recovering the resistivity of the uninvaded portion of a single layer. The response from unwanted parts of the formation is lumped under the term "environmental effects."
The main objectives of resistivity logs are the determination of Rt and Rxo and, for the newer imaging devices, the mapping of resistivity profiles into and around the borehole. This is accomplished by incorporating devices with at least three, and preferably more, depths of investigation. The deep-reading focused devices include the deep laterolog device (LLd) and the deep induction (ILd) device. Medium-depth devices include the shallow laterolog (LLs) and the medium induction (ILm). The shallow devices include the microresistivity device or the spherically focused log (SFL). The latest laterolog and induction tools include arrays focused at many depths of investigation—from five to eight depths—measured simultaneously. These allow a better description of the invaded zone and allow for interpretation of complex invasion profiles.
Electrode Resistivity Devices
Normal and Lateral DevicesDuring the first quarter century of well logging, the only electrical surveys (ES) available were the resistivity logs made with so-called lateral and normal devices plus the SP. Thousands of them were run each year all over the world. Since then, new logging methods have been developed to measure values much closer to Rxo and Rt. Nevertheless, the conventional ES logs (consisting of SP; 16-in. normal; 64-in. normal; and 18-ft, 8-in. lateral) are stored in log archives all over the world. Because new information can often be obtained by reinterpreting old ES logs, this chapter includes discussion of the principles and responses of the ES measurements.
The first resistivity devices were the normals and laterals. These were, in concept, extensions to laboratory four-terminal resistivity-measuring cells. Current is injected in the formation from a single electrode and returned to a point remote from the well. The current near the injection electrode spread out radially from the electrode. Two voltage–measuring electrodes (M and N) on the sonde approximated the measurement of a constant-voltage spherical shell around the injection electrode. The measurements of voltage and current are converted to a resistivity measurement.
For normal devices (Fig. 3B.2), the distance AM is small: 1 to 6 ft as compared with MN, MB, and BN. In practice, N or B may be placed in the hole at a large distance above A and M [the voltage measured is practically the potential of M (because of current from A), referred to an infinitely distant point]. The distance AM of a normal device is its spacing. The point of measurement is midway between A and M. The most common normal spacings were 16 and 64 in.
Interpretation of laterals and normals is very complicated because the response is a complicated function of the formation being measured. Fig. 3B.4 shows a computed response of the 16- and 64-in. normals for a series of beds with and without invasion. The separation is not a clear function of invasion, but is also a function of bed thickness. Fig. 3B.5 shows the 18-ft, 8-in. lateral tool in the same series of beds. The relation of the curve to the bed is not clear at all. Many charts (called departure curves) were published to aid in interpretation of the ES logs. Modern interpretation methods include 2D inversion (after the curves are digitized) and iterative forward modeling for when they are not digitized.
LaterologsLaterolog devices are designed to minimize the influence of near-tool effects (i.e., the borehole and the invaded zone) for a deep laterolog measurement, as well as to reduce the response from adjacent beds.
To estimate Rt under a variety of different logging conditions and in different formations, a simple three-parameter, step-profile invasion model is often used. This model consists of a flushed zone of resistivity Rxo and a sharp boundary at diameter di, with the uninvaded zone of resistivity Rt. Three independent, borehole-corrected resistivity measurements with appropriately chosen depths of investigation contain enough information from the formation to reliably solve for Rt using this model. Measurements with the following features should be chosen: small, correctable borehole effects; similar vertical resolutions; and well-distributed radial depths of investigation—one reading as deep as practical, one very shallow reading, and one intermediate reading.
In conductive muds, the Dual Laterolog (DLL) Resistivity– Rxo combination tool provides simultaneous measurements suitable for evaluating Rt, Rxo, and di. It should be said that the value of Rt in a given bed is an interpreted parameter, and is almost never measured. As long as the formation is invaded, assumptions about the invasion profile must be made to estimate Rt.
Dual Laterolog Resistivity Measurements. Fig. 3B.6 shows the electrode array used for deep and shallow laterolog measurements (LLd and LLs, respectively). Both logs share the same electrodes and have the same current-beam thickness, but different focusing currents give them different depths of investigation. The measure current (I0) is emitted from the central A0 electrode, returning to an "infinitely distant" electrode, usually at the surface. The focusing current (Ia) flows from the A1 and A2 (and A1′ and A2′) electrodes to a distant electrode for the LLd measurement and from A1 to A2 (and A1′ to A2′) for the LLs measurement. The focusing current is adjusted so that electrodes M1 and M2 (and M1′ and M2′) remain at the same potential.
A constant-power measuring system ensures measurement accuracy over a wide range of resistivities (from 0.2 to 40,000 ohm•m). Both the measure current and measure voltage (V0) are varied and measured, but their product power I0V0 is kept constant.
Long guard electrodes are required to achieve the desired depth of investigation and measurement range. For electrode tools focused using guard electrodes, the depth of investigation increases only as the square root of the length of the guard electrodes. This requirement results in the 28-ft [8.5-m] total length of the LLd electrode array. A beam thickness of only 28 in. [0.7 m], however, ensures good vertical resolution.
The LLs measurement shares most of the electrodes with the deeper measurement. This is achieved by operating LLs and LLd at different frequencies. The LLs and LLd measurements have the same vertical resolution, but the LLs device uses a less constrained focusing condition in which the focusing current returns to electrodes on the array instead of to a remote electrode. The LLs measurement therefore has a shallower depth of investigation and responds more strongly to the region around the borehole that is usually affected by invasion.
Laterolog Anomalies. The Groningen effect was named after the large Dutch gas field where the anomaly was first identified. The effect is an anomalously high resistivity reading that occurs for approximately 100 ft [30 m] below a thick, highly resistive bed such as the thick evaporitic Zechstein caprock at the Groningen field. The effect is maximum around 1 ohm•m. Because the DLL measure current is AC (albeit very low frequency), skin effect reduces the volume around the well where the measure and focusing current can flow. Little of the current is able to return to the remote electrodes through the highly resistive formation, with the majority flowing in the conductive mud in the borehole. This creates a negative potential at the far reference electrode used as the potential reference for the laterolog measurement. If casing has been set in or below the resistive zone, it accentuates the "short circuit" effect of the borehole, and the Groningen effect is more pronounced. Drillpipe conveyance produces the same effect, with the drillpipe becoming the "short circuit." This problem severely limits the use of drillpipe conveyance of the DLL in high-angle or horizontal wells in many reservoirs.
A mild Groningen effect may be difficult to identify from the LLd curve alone. The Schlumberger DLL has a modified-geometry measurement that can also be recorded. This provides an LLg curve that separates from the LLd curve when Groningen effect is present. If the Groningen effect is positively identified, an estimate of its magnitude can be made by analyzing the signal phases in the tool, and an approximate correction can be applied to the log. The LLs measurement uses different current paths and does not suffer from the Groningen effect. The array laterolog (see the following) is not affected by Groningen effect.
Azimuthal Dual Laterologs. In the early 1990s, a new dual laterolog that had an additional azimuthally segmented current electrode was introduced. The Schlumberger ARI* Azimuthal Resistivity Imager has a set of segmented azimuthal electrodes incorporated in a conventional dual laterolog array. The tool records azimuthal resistivity variations around the borehole and produces an image of the variations. The azimuthal electrodes are placed at the center of the A2 electrode of the DLL tool and do not interfere with the standard LLd and LLs measurements.
The deep azimuthal measurement operates at the same frequency as the deep laterolog measurement, and the currents flow from 12 azimuthal current electrodes to the surface. They are focused by the current from the A2 electrode’s upper and lower portions and by currents from the other current electrodes. In addition, the current from each azimuthal electrode is focused passively by the currents from its neighbors. The resulting operation of the azimuthal array has no effect on the LLd and LLs measurements.
Twelve azimuthal resistivities are computed, and from their sum, a high-resolution resistivity measurement, LLhr, is derived. This is equivalent to replacing the azimuthal electrodes with a single cylindrical electrode of the same height.
The high-resolution LLhr curve reads almost as deeply into the formation as a deep laterolog LLd curve, particularly when Ryo is less than Rt. An LLhr log can therefore replace an LLd log for interpretation, especially where its vertical resolution is an advantage. Individually selected azimuthal resistivities can be used in the same manner where the logged interval is azimuthally anisotropic or includes highly dipping thin beds.
The azimuthal resistivity measurements are sensitive to tool eccentering in the borehole and to irregular borehole shape. Auxiliary measurements are made that are very shallow, with current paths close to the tool. Most of the current returns to the A2 electrode near the azimuthal array. Because the borehole is generally more conductive than the formation, the current tends to stay in the mud, and the measurement responds primarily to the volume of mud in front of each azimuthal electrode. The measurement is therefore sensitive to borehole size and shape and to eccentering of the tool in the borehole. However, for best use of the azimuthal measurements, the tool should be well centralized in the borehole.
A High-Resolution Azimuthal Laterolog Sonde (HALS) resembles an ARI or a dual laterolog array. Although it is shorter than either of these tools, the HALS is not just a scaled-down version. Its dimensions are optimized to achieve similar performance as an ARI sonde with a tool that is only approximately one-half its overall length.
Like the ARI tool, the azimuthal array of the HALS makes deep and shallow resistivity measurements around the borehole with a 1- or 2-ft [0.3- or 0.6-m] vertical resolution. (In this chapter, vertical resolution is defined as the 90% width of the vertical response function.) Formation resistivity images can be derived from either the deep or the shallow measurements. Mud resistivity and tool standoff are also measured. In addition to providing a visual image of formation lamination and anisotropy, the azimuthal images can be used to estimate the gross formation dip and to correct deep resistivity measurements in dipping beds.
Because the HALS is shorter than ARI and DLL sondes, the borehole effect of the shallow measurement is larger. Combined with the slightly reduced depth of investigation of the deep array, this reduces the precision of the invasion correction in cases of invasion where di > 50 in. [1.3 m]. The inherent vertical resolution is sharper—24 in. [0.6 m] for the HALS deep and shallow resistivity curves (HLLD and HLLS, respectively) compared with 40 in. [1 m] for the LLd and LLs curves of the DLL log. The HALS provides high-resolution deep and shallow curves (HRLd and HRLs, respectively) with the same 12-in. vertical resolution as the LLhr curve of an ARI log.
Real-time corrections can be made for Groningen effect, electrical path changes imposed by tough logging conditions (TLC) logging in which the logging tool is transported on drillpipe, and borehole effects. A two-parameter inversion model can also be used in real time to solve for Rt and di, with Rxo provided by the microcylindrically focused log (MCFL) measurements of the Platform Express tool.
* Throughout this chapter, tool names are service marks of the referenced companies.
Array Electrode ToolsThe most recent development in electrode tools is the array laterolog or array lateral tools. These combine multiple depths of investigation with 2D inversion of the data to give much improved response in invaded thin beds with conductive mud.
The Schlumberger high-resolution laterolog array (HRLA) tool consists of five laterolog arrays with different depths of investigation. All current is returned to electrodes above and below the array, so no bridle is used. Because it has no bridle, it does not suffer from Groningen effect.
Borehole and shoulder effects are minimized by the use of laterolog-style focusing. Focusing involves injecting current from guard or bucking electrodes to ensure that the current from the central measure electrode flows into the formation rather than along the borehole. By having all currents return to the tool body rather than surface, Groningen effect is eliminated and shoulder-bed effect reduced. More importantly, the surface current return and insulating bridle are no longer needed. All signals are measured at the same time and logging position. This avoids horns or oscillations caused by irregular tool motion and ensures that the measurements are always exactly depth-aligned.
The HRLA tool uses segmented bucking electrodes and multifrequency operation (ranging from 75 to 270 Hz) to acquire six simultaneous measurements. The six modes are focused by a combination of hardware and software focusing. The hardware injects the currents in a way that is as close to focused as possible. The shallowest mode, RLA0, is mostly sensitive to the borehole and is used to estimate the mud resistivity. The apparent resistivities RLA1 through RLA5 are all sensitive to the formation, becoming progressively deeper in investigation. Fig. 3B.7 shows the radial response of the optimized HRLA tool compared to the HLLd and HLLs measurements from the HALS tool. Fig. 3B.8 shows the HRLA logs compared to the DLL, LLd, and LLs logs.
The inversion process is initiated by using the shallow measurements (associated with short-spacing sensors) to identify and evaluate the shallow-formation resistivity structure. Deep measurements (associated with the longest sensor spacings) are used in evaluating the uncontaminated formation resistivity structure. The intermediate measurements are used to derive the radial-invasion profile. Using inversion processing, measurements of different vertical resolution and depths of investigation are combined in a single interpretation process to provide an accurate resistivity distribution image.
Final inversion results consist of the estimated resistivity structure satisfying all HDLL data and the statistical quality indicators represented in terms of the importance of the formation parameter along with its corresponding error bounds. Fig. 3B.10 shows a comparison of HDLL Rt and Rxo inversion results with conventional Dual Laterolog (RS is DLL shallow and RD is DLL deep) and Micro Laterolog (MLL) measurements.
The SFL device measures the conductivity of the formation near the borehole. It uses small electrodes that can be combined with the dual-induction tool to provide shallow-investigation data for invasion evaluation.
The SFL device uses two independent current systems. A focusing current system establishes constant-potential spherical "shells" around the current electrode, even in the presence of a conductive borehole, and the I0 survey current flows through the volume of investigation.
The SFL electrode array consists of current-emitting, current-return, and measure electrodes. Two equipotential spheres are established around the I0 survey current electrode; the first sphere is approximately 9 in. [0.2 m] from the electrode, and the other is approximately 50 in. [1.3 m] from it. The volume of formation between these two surfaces is constant, and because a potential difference of 2.5 mV is maintained between the spheres, the conductivity of this volume of formation is proportional to the I0 survey current intensity.
Cased-Hole Resistivity ToolsDespite the apparent paradox of measuring formation resistivity through the highly-conductive steel casing, tools are now available that can measure the formation resistivity to considerable accuracy. The idea originated in the 1930s and was revisited by Kaufman and Vail in the late 1980s. Commercial tools were introduced in 2000 by Schlumberger and Baker Atlas. Both of these tools operate on the Kaufman-Vail principles.
The Schlumberger cased-hole formation resistivity (CHFR) tool has three sets of four arms that contain electrodes that are forced into contact with the inside of the casing. A current generator on the surface is connected to an electrode at the top of the tool. The current is injected into the casing and returns to an electrode in the earth some distance from the casing. Although most of the current returns through the casing, some small fraction of it will leak off from the outside of the casing and will return through the earth. This leakoff current forms the basis for the CHFR measurement.
The leakoff current is determined by measuring the voltage drop along a section of the casing. The double-differenced voltage Δ contains both the leakoff term and the voltage drop produced by the current flowing through the casing and the resistance, R, of the casing. The current switch is changed to position 2. Now a current from inside the tool is sent from the upper electrode to a lower electrode. The voltage difference is now measuring the resistance of the casing, R1. All of the measurements are combined in the equation
to produce a formation resistivity measurement. The measurements are taken while the tool is stationary and take approximately a minute per station.
Fig. 3B.11 shows a log of the CHFR in a newly cased well compared with open-hole HALS and AIT logs. The comparison is very good, and in the zone from 865 to 900 ft, with Rxo > Rt, the CHFR agrees well with the AIT 90-in. log, showing the great depth of investigation of the CHFR log.
The Baker Atlas Through-Casing Resistivity (TCR) tool operates on a similar measurement principle as the CHFR. It is smaller in diameter (2 1/8 vs. 3 1/8 in. for the CHFR).
GVR Resistivity-at-the-Bit Tool. The Schlumberger Geovision Resistivity (GVR) tool is an electrode resistivity tool that measures five resistivity values—bit, ring, and three button resistivities—as well as gamma-ray and shock measurements.
A 1500-Hz alternating current passes through the toroidal-coil lower transmitter that is 1 ft from the bottom of the tool, inducing a voltage in the collar below. Current flows through the collar and bit and into the formation in front of the bit, returning to the collar farther up the drillstring. The resistivity at the bit is derived from the axial current, which is measured by a ring monitor toroid, and the induced voltage, which is a function of the transmitter current.
When the GVR tool is positioned directly above the bit, the resistivity measurement has a resolution of approximately 2 ft [61 cm], which is usually adequate for "geostopping"—stopping drilling precisely at casing or coring depths.
Focused Multidepth Resistivity. There are four focused-resistivity measurements incorporated in the RAB tool. These include the ring electrode measurement, with a depth of investigation of approximately 9 in. [0.23 m], and three button electrode measurements, with depths of approximately 1, 3, and 5 in. [2.5, 8, and 13 cm, respectively] into the formation.
The button measurements are radial, acquiring azimuthal resistivity profiles as the tool rotates in the borehole. A rotational speed of at least 30 rpm is required for full profile recording, with each button recording 56 resistivity measurements per rotation. The data are usually stored downhole for later retrieval, although a compressed image and selected button data may be transmitted to the surface in real time together with the ring and bit resistivities and gamma-ray measurements. Fig. 3B.12 shows a recorded GVR image compared with an image from the wireline FMI borehole resistivity image tool (see the following for a description of this tool).
All four focused resistivities use the same measurement principle: Current from the upper transmitter flows down the collar and out into the formation, leaving the collar perpendicular to its surface and returning to the collar above the transmitter. Low-impedance circuits measure the current at each button electrode, and the axial current flowing down the collar is measured at the ring electrode by the ring monitor toroid and at the lower transmitter by the lower monitor toroid. These resistivity measurements are repeated using current from the lower transmitter.
In a homogeneous formation, the equipotential surfaces near the button and ring electrodes on the RAB tool are cylindrical. However, in layered formations, there is a tendency for current to flow preferentially in the more conductive beds and avoid the more resistive beds. This effect is known as "squeeze" for conductive beds and "antisqueeze" for resistive beds, and it leads to horn-like distortion of resistivity readings at bed boundaries.
A cylindrical focusing technique (CFT) is used to measure and compensate for this distortion by restoring the cylindrical geometry of the equipotential surfaces in front of the measurement electrodes. This is achieved by regulating the currents generated by the upper and lower transmitters for zero axial current flow at the ring monitor electrode, which avoids current flow along the borehole and focuses the ring current into the formation. This focusing technique produces a response very similar to that of a wireline laterolog.
Environmental Effects on Laterolog ToolsLaterolog and SFL log readings are influenced by the borehole mud, adjacent shoulder beds, and the invaded zone as well as the uninvaded formation. If automatic corrections are not available, log-interpretation charts provided by the service company are used to manually correct the log readings for these influences. The borehole corrections must always be made first, followed by bed-thickness corrections and finally invasion corrections of the determination of Rt, Rxo, and di.
Invasion Corrections. The "geometric factor" relates the effect of a portion of formation on the logging tool reading to its position relative to the tool in an infinite homogeneous medium. It has a particular application to induction logging tools, but pseudogeometrical factors are a useful comparative tool for other resistivity devices.
Fig. 3B.13 is a plot of integrated pseudogeometrical factors for several focused resistivity logs. It graphically compares the relative contributions of the invaded zone to the tool responses and their relative depths of investigation. The good spread in radial characteristics of the LLd and LLs measurements enables accurate resistivity analysis over a wide range of invasion conditions.
To evaluate the three unknowns of the simple step-profile invasion model (Rxo, Rt, and di), a combination of at least three carefully chosen resistivity measurements is required. LLd and LLs curves, with a very shallow resistivity measurement that reads Rxo directly, may be sufficient. See the section on invasion interpretation for more details on the determination of Rt, Rxo, and di.
Induction logging was originally developed to measure formation resistivities in boreholes containing oil-based muds and in air-drilled boreholes because electrode devices could not work in these nonconductive boreholes. However, because the tools were easy to run and required much less in the way of chart corrections than laterals or normals, induction tools were used in a wide range of borehole salinity soon after their introduction.
PrinciplesCommercial induction tools consist of multiple coil arrays designed to optimize vertical resolution and depth of investigation. However, to illustrate induction-tool fundamentals, it is instructive to first examine the basic building block of multiple-coil arrays, the two-coil sonde.
Fig. 3B.14 shows that a two-coil sonde consists of a transmitter and receiver mounted coaxially on a mandrel. Typical coil separations range from 1 to 10 ft apart. In practice, each coil can consist of from several to 100 or more turns, with the exact number of turns determined by engineering considerations. The operating frequency of commercial induction tools is in the tens to hundreds of kilohertz range, with 20 kHz being the most commonly used frequency before 1990.
The induction transmitter coil is driven by an alternating current that creates a primary magnetic field around the transmitter coil. The primary magnetic field causes eddy currents to flow in a continuous circular distribution (often mistakenly called "ground loops") centered around the borehole axis. The color contours in Fig. 3B.14 show the current distribution. These eddy currents are proportional to the formation conductivity, and they in turn generate a secondary magnetic field, which induces an alternating voltage in the receiver coil. This receiver voltage is first-order proportional to the conductivity of the formation.
Because the transmitter current is alternating, there is a phase shift between the transmitter current and the current density in the formation. This phase shift is not the same in all parts of the formation—it increases with distance into the formation (Fig. 3B.14). Similarly, the phase in the receiver is even further shifted. At very low conductivities, the total phase shift is approximately 180° and increases with increasing formation conductivity. Induction tools have always measured the part of the voltage that is exactly 180° phase-shifted from the transmitter current (called the R-signal). As the conductivity increased, and the phase shifted, the voltage was a bit less than expected from a linear relationship. This difference is called skin effect. Modern induction tools make an additional measurement at a phase shift of 270° from the transmitter current (called the X-signal). These two measurements, being in quadrature, allow precise phase and amplitude measurement of the receiver voltage.
To produce adequate sensitivity to the uninvaded zone, induction tools perforce include signals from a large volume of formation. The challenge is to determine exactly where the measurement is coming from in the formation. Doll defined the geometrical factor as a 2D function g(ρ,z), which defines the part of the total signal that comes from an infinitesimally thin loop around the borehole. This definition is valid only at very low conductivities. Moran defined a modification of the geometrical factor that is valid in low contrast formations at any conductivity. This response is known as the Born response.
The response to formation layers is given by the vertical response function gV(z), which is defined as the integral of the 2D response function g(ρ,z) over radius ρ. The response to radial variations in a thick bed is given by the radial response function gR(ρ), which is defined as the integral of g(ρ,z) over z. The response of the array to invasion in a thick bed is characterized by the integrated radial response GR, which is the cumulative integral of gR(ρ) over radius.
Multicoil ArraysBecause the direct transmitter-receiver mutual coupling of a two-coil array can produce a voltage several thousand times that from a formation, two-coil arrays are not practical. The simplest practical array is a three-coil array with a transmitter and two receivers. The second receiver is placed between the transmitter and main receiver, and is wound oppositely so that the voltages in the two receivers exactly cancel when the array is in free space. The response is the sum of the coil-pair responses.
One of the most successful induction arrays was the 6FF40 array introduced in 1960. It had three transmitters and three receivers, with a symmetric Born response g. Figs. 3B.15 and 3B.16 show its vertical and radial responses. The array was designed to achieve deep investigation, reasonable vertical resolution, and a low borehole effect. However, the large peaks in the 2D response along the tool resulted in sensitivity to borehole washouts, called cave effect.
Dual-Induction ToolsOne of the challenges of measuring formation resistivity is to sort out the resistivity of the invaded zone from that of the virgin zone. The earliest concept to successfully solve the problem (at least in thick beds with uncomplicated invasion profiles) was the dual-induction tool. This tool combined a 6FF40 array as a deep-induction measurement (ID or ILD) with a set of receivers that worked with the 6FF40 transmitters to produce a shallower measurement. This was referred to as the medium-depth induction (IM or ILm).
Because there are three parameters in the simplest step-profile invasion model, at least three measurements are required to solve for these parameters. The shallow measurement was a shallow laterolog (LL8 or SFL) colocated with the induction arrays. The radial response function involves very complicated mathematics, and the solution offered to users of the dual induction logs was a graphical solver called the tornado chart.
he ILD-ILM-SFL logs separate when there is invasion, and this separation is what allows interpretation for invasion parameters. Fig 3B.17 shows the modeled response of the dual induction-SFL tool (DIT) in a typical Gulf of Mexico pay zone with a transition over a water zone.
Some sort of function must be applied to the tool voltages to correct for this nonlinearity. The processing applied to the Schlumberger DIT consisted of a skin-effect function ("boost") applied to the measured R-signals from the induction arrays. This was based on computations of the response in an infinite homogeneous medium. The ILd was further processed using a three-station deconvolution filter to slightly sharpen the bed-boundary transition and to correct for shoulder effect over a limited resistivity range (1 to 10 ohm•m). At other formation-resistivity ranges, the response either produced horns or large shoulder effects. Fig. 3B.18 shows the DIT logs in a set of formation layers with the same shoulder-bed contrasts, but centered on 1, 10, and 100 ohm•m.
Borehole correction was also hand-applied to the induction and SFL logs. The borehole correction chart was derived from measurements made with a DIT in plastic pipes full of salt water. The 6FF40-based dual induction-shallow electrode tool was offered by most service companies.
Phasor InductionThe DIT tool became the standard resistivity tool and remained virtually unchanged for more than 20 years. However, as its application moved from the original Gulf of Mexico formation contrasts to higher-resistivity formations, the shoulder-effect problem became much worse. Although shoulder-correction charts were provided for high resistivity, they mainly indicated that the problem was bad rather than serving as a usable correction mechanism.
The fundamental problem in induction log interpretation is to isolate the response of a thin bed and the virgin zone from the shoulders and the invaded zone after the measurement process has thoroughly mixed them. The Phasor induction tool was introduced in the mid-1980s and was the first tool to automate the environmental corrections. It uses a linear deconvolution function to correct for shoulder effect and uses the X-signal measurement to correct for skin effect. This algorithm was the basis for Phasor Processing. It can be shown that a filter fitted at low conductivity works well at low conductivity but produces large errors at high conductivity. The error is, however, a slowly varying function closely related to the X-signal. An algorithm applied to the X-signal to match it to the skin-effect error allows a single FIR filter to correct for shoulder effect over a wide range of conductivities.
Fig. 3B.19 shows the results of Phasor processing in the formation models of Fig. 3B.18. The induction logs are fully shoulder-effect-corrected at all conductivity levels. Phasor logs in the Gulf of Mexico simulation of Fig. 3B.17 are not very different from the DIT logs. This is in part because this formation is where the DIT logs were designed to work well. Although tornado charts were published for the Phasor induction logs, the invasion parameters are computed in real time at the wellsite. Borehole corrections are based on computer models of an eccentered tool in a wide range of borehole salinities and formation conductivities. Borehole corrections are applied in real time at the wellsite. The Phasor induction tool was the first induction tool that could provide full environmental correction and invasion parameter determination at the wellsite. In 1987, changes to the deconvolution filters allowed induction logs with a 2-ft vertical resolution (compared with 5 ft for ILm and 8 ft for ILd).
However, all of these tools are based on two induction arrays—a shallow array and a deep array. Performance in complex invasion profiles is limited by the small number of measurements. Fig. 3B.20 shows the Phasor logs in a simulation taken from a field log in a gas reservoir. Here an annulus has developed, and the deep log reads much less than Rt. In this well, the three-parameter invasion model will not return the correct value of Rt.
In the case of oil-based mud (OBM), the SFL is not usable. Separation between the medium and deep logs is only a qualitative indication of invasion and is not quantitatively interpretable.
Array-Induction ToolsWith the Phasor Induction tool, the dual-induction concept had reached its limits. In particular, improvements were needed in better estimates of Rt in the presence of deep-invasion or complex transition zones. As the grosser environmental distortions were corrected by Phasor Processing or similar processing, annulus profiles and other transitions were encountered more often.
These response problems, coupled with an increasing use of OBM, led to the concept of using several induction arrays with different depths of investigation. With the problems of applying linear deconvolution filters solved, then Doll’s approach of using a simple array was applicable. The Schlumberger AIT was designed with eight simple three-coil arrays ranging in length from 6 in. to 6 ft.
Array-Induction Principles. The first step in log formation in the AIT family of tools is to correct all raw array signals for borehole effects. This process is based on a forward model of the arrays in a circular borehole, and it includes an exact description of the tool in the model.
The signal measured by an induction sonde eccentered in a borehole can be described mathematically as a function of four parameters. These are the borehole radius r, the mud conductivity σm, the formation conductivity σf, and the tool position x with respect to the borehole wall (commonly referred to as the "standoff").
The correction algorithm is designed to solve for some of these parameters by minimizing the difference between the modeled and actual logs from the four shortest arrays. The information content of these measurements is not sufficient to solve for all the borehole parameters at the same time. In practice, two of the four parameters can be reliably determined by this method. The other two parameters have to be either measured or fixed. The equivalent homogeneous formation conductivity σf must always be solved for because no measurement is closely enough related to it. This leaves one of the other parameters to be determined, and the remaining two parameters must be entered as measurements. This leads to the three borehole correction methods to compute mud resistivity, hole diameter, and standoff. All of the AITs except the original AIT-B have integral mud resistivity sensors, and "compute standoff" is the default borehole-correction method in water-based mud (WBM).
A method was developed to combine these array measurements to focus the resulting log to the desired depth of investigation, while at the same time doing so with a high vertical resolution and minimizing cave effect. The log-formation process is described by the equation
In this equation, σlog is the recorded AIT log, σa(n) is the measured log from the nth channel, and N is the total number of measure channels. This process produces a log that is different from that produced by any of the individual arrays. It is still characterized by a response function. This response function is a weighted sum of the response functions of each of the individual channels n. Skin effect is handled in a manner similar to the Phasor tool.
The result of this equation is a combination of the logs from the eight array that "distills" the radial information from the eight arrays into five independent logs with depths of investigation of 10, 20, 30, 60, and 90 in. Each of these five logs is available at a resolution of 1, 2, and 4 ft. The radial profile is identical at all resolutions, and the vertical resolution is identical for all radial depths. The set of weights w in Eq. 3B.1 determines which log is produced.
If the mud is very salty, or if the borehole is very large, the signal in the AIT arrays from the borehole will be very large. With salty mud, even normal variations in the borehole surface from the drilling operation can cause "wiggles" on the short array data. Several years of practice have shown that these can affect the final logs, especially the 1-ft logs. This experience has shown that in an 8-in. borehole with 1 1/2-in. standoffs, the 1-ft logs are normally usable at Rt/Rm contrasts up to 100; the 2-ft logs are usable up to a contrast of 450, and the 4-ft logs are usable up to contrasts of 1000. Algorithms based on the real-time use of a chart and "road-noise" analysis of the 6-in. array allow real-time selection of the appropriate resolution based on actual logging conditions.
AIT logs separate in invaded zones and give a good visual indication of invasion, even with OBM. Fig. 3B.21 shows the AIT logs in the same formation as Fig. 3B.17 (left). In the annulus case, Fig. 3B.21 (right), the AIT logs are "out of order," clearly indicating the nonstep nature of the invasion profile.
Interpretation of logs in deviated wells or where the apparent dip is high is considerably complicated. First, one has to recognize that the logs are at high apparent dip. Fig. 3B.22 shows AIT logs in a formation with an apparent dip of 85°. Although this is high, the characteristics that appear here—horns and strange log order—appear in logs with dips as low as 40°. In many fields, faults and slumping of young sediments can produce high apparent dips that are not detectable on seismic profiles. Merlin processing has been developed to produce logs fully corrected for dip effect. Recently, real-time high-angle processing was made available. This processing produces logs that are independent of dip angle. However, the resulting logs are also shallow. Fig. 3B.23 shows dip-invariant processing (Grimaldi) on the right.
Users of induction logs should be very careful making quantitative analyses in wells that are deviated, or if the formation is dipping. If the shoulder-bed contrast is 20 or less, then the minimum angle where dip correction is needed is approximately 30°. At shoulder-bed contrast of over 100, the logs will need correction at dips as low as 10°.
Field-Log ExamplesA few field-log examples will illustrate the richness of information available in array-induction logs. The first example, Fig. 3B.24, is a comparison of AIT and Phasor induction logs in a gas zone from Canada. The AIT shows a nonmonotonic curve order, indicating an annulus profile. If the data from this zone is inverted into an annulus profile using material-balance constraints to determine the thickness of the annulus, then the complete annulus parameters can be recovered (Fig. 3B.25).
Other Array Induction of ToolsBaker Atlas introduced its High-Definition Induction Log (HDIL) array induction tool in 1996. It is a seven-array tool that operates at eight frequencies. This information can be processed in a variety of ways, depending on the environment. The multiple frequencies are used for skin-effect correction. This algorithm is developed by computing the R signal measured by a given array at each frequency at a wide range of formation conductivities. The data at each frequency are fitted to the true formation conductivity. The resulting function is used for the skin-effect correction.
The skin-effect-corrected conductivities are then deconvolved with filters to form six logs at depths of 10, 20, 30, 60, 90, and 120 in., and at three matched resolutions of 1, 2, and 4 ft. An additional presentation is the "true resolution" log set. This has the same six depths, but the resolution of each depends on the depth of investigation. This presents the resolution information content that actually comes from the formation region near the midpoint of the integrated radial response function. Dip correction is provided at the computing center. Fig. 3B.28 shows an example. Fig. 3B.29 shows a 2D inversion available at the computing center.
Halliburton introduced its High-Resolution Array Induction (HRAI) tool in 2000. It is a six-array tool based on the array layout of the HRI. Standard HRAI tool logs present resistivities at vertical resolutions of 4, 2, and 1 ft, each with six depths of investigation (10, 20, 30, 60, 90, and 120 in.). The log resistivities are inverted to yield true resistivities of the formation in the virgin zone, Rt, and in the invaded zone, Rxo, near the borehole. Invasion diameters (Di) corresponding with Rt and Rxo are also presented. HRAI tool answer products are available in real time while logging.
A variant on the array induction principle was introduced by Weatherlord (previously Reeves Wireline). The array induction donde (AIS) combines four simple induction arrays with a shallow focused-electrode array. The induction data have been presented in two ways. Originally, the four arrays were combined in software to match the response of the ILd and ILm arrays. Later, the Vectar processing was introduced to produce a higher-resolution log. Data from each array is skin-effect-corrected and then resolution-matched to the shortest array. Up to six curves are presented from the four arrays.
LWD Induction Tools
Induction Tools on the Drillpipe
Commercial resistivity measurements made while drilling first became available in the late 1970s. Because the drilling environment is much more adverse than the wireline logging environment, a simple short normal tool mounted behind the drill bit was used as the first LWD resistivity tool. However, short normal tools were only able to provide information for basic interpretation, such as correlation of geological markers and estimation of gross water saturation, because of their shallow depth of investigation and relatively poor vertical resolution. Being DC electrode devices, normal tools are also limited to conductive mud environments.
To expand the LWD resistivity market to OBM environments, induction-type propagation measurements were introduced in the early 1980s. The first commercial device was the electromagnetic wave resistivity (EWR) tool from NL Information Services (later merged with Sperry-Sun). Shortly after this, Schlumberger introduced the compensated dual resistivity (CDR) tool. All LWD propagation tools are run with a gamma ray tool for lithology estimation and correlation. Log data are transmitted uphole in real time using mud-pulse telemetry. Downhole memory and batteries allow raw and processed data to be stored for later retrieval.
Principles of the Propagation MeasurementBecause the CDR is a simple tool that measures both attenuation and phase shift, it is used to demonstrate the basic concepts of propagation measurements. Conventional induction measurements are made with mutually balanced arrays of transmitter and receiver coils operating in the kilohertz frequency range and that phase-lock measurement electronics. Because it was difficult to engineer this type of measurement on a steel drill collar using the technology of the early 1980s, a higher-frequency propagation measurement was considered to be more practical for LWD. A frequency of 2 MHz was chosen because it was the lowest frequency at which accurate propagation measurements could be made on a drill collar at that time.
The CDR tool broadcasts a 2-MHz electromagnetic wave. A propagation measurement is made by taking the difference between the phases (phase shift) and amplitudes (attenuation) of the voltages recorded at the two receivers. Attenuation increases as a function of increasing conductivity, while the wavelength decreases as conductivity increases. Thus, the two measurements are proportional to formation conductivity and can be used to generate resistivity logs. Note that the CDR tool has two transmitters. The phase-shift and attenuation measurements generated by the upper transmitter between the two receivers, and by the lower transmitter between the two receivers, are averaged to symmetrize the tool response. This averaging is known as borehole compensation because it also reduces the effect of borehole rugosity. The transmitter-to-receiver spacings for the CDR tool are 25 and 31 in.
The phase-shift and attenuation measurements are transformed to two independent resistivities, which for the CDR tool are known as RPS (phase shift, shallow) and RAD (attenuation, deep). Because at 2 MHz dielectric effect can be significant at high resistivity levels, a dielectric correction is performed before the raw data are converted to apparent resistivity. Most service companies have developed their own proprietary algorithms to perform dielectric correction. Joint inversion for both resistivity and dielectric constant is also possible with today’s multiarray propagation tools. The dielectric-corrected phase shift and attenuation are converted to resistivity using a table look-up algorithm based on computed tool response in homogeneous isotropic media of known resistivity,Rt.
The phase-shift and attenuation measurements are both relatively insensitive to borehole size and mud resistivity. Borehole correction is only necessary in conductive holes with large washouts when the Rt/Rm contrast is greater than 100 to 1.
Invasion is usually quite shallow at the time of drilling when most LWD logs are run. However, LWD logs may also be recorded each time that the drillstring is pulled to replace the drill bit. At these later times, invasion can be much deeper. The two resistivity values, RPS and RAD, provide two independent depths of investigation as an indication of invasion.
The reason that two depths of investigation can be obtained from a single measurement is made clearer by examining the behavior of the electromagnetic field. The surfaces of constant phase are spheres because the wave travels with the same speed in all directions. The surfaces of constant amplitude are toroids because the wave is stronger in the radial direction than in the vertical direction, which is a normal characteristic of vertical magnetic dipole antennas. If we compare the phase and amplitude contour lines passing through the receivers, the amplitude extends to a significantly deeper region of the formation than the phase.
Depth of investigation can also be studied by modeling tool response in invaded formations. Fig. 3B.30 shows CDR radial response for a case with Rxo > Rt, and Fig. 3B.31 shows the radial response for Rxo < Rt. In both figures, RPS and RAD are plotted as a function of increasing invasion radius. In Fig. 3B.30, RPS reads consistently closer to Rxo, indicating that RPS is the shallower of the two measurements. In Fig. 3B.31, RPS is again consistently shallower than RAD. In this case, the RPS curve extends below the value of Rxo between radii of 30 and 50 in. because of wave reflection at the invasion front. In general, the depth of investigation of RPS is 10 to 20 in. shallower than that of RAD.
Vertical resolution is characterized in more detail in Fig. 3B.32, which compares CDR and Phasor induction logs. In this low-resistivity formation, the vertical resolution of CDR log is slightly sharper than that of the Phasor log. (The SFL log indicates that shallow invasion has taken place at wireline time.) However, the vertical resolution of 2-MHz logs deteriorates as the formation resistivity level increases. Because conventional induction logs always undergo vertical processing, while 2-MHz logs are seldom processed because their depth sampling is irregular, care must be taken when comparing wireline induction and 2-Mhz logs in resistive formations. Fig. 3B.33 shows the variation in CDR vertical resolution as a function of resistivity level.
Multiarray Propagation ToolsDuring the 1990s, all major service companies developed multiarray versions of 2-MHz tools. Schlumberger introduced the array resistivity compensated tool (ARC5) 46 as a replacement for the CDR tool and to accommodate the increasing number of slimholes being drilled. The ARC5 tool makes five independent phase-shift and attenuation measurements. The number of measurements was deliberately chosen to be the same as that of wireline array-induction tools to allow the sharing of interpretation methods for analyzing complex invasion profiles and estimating Rt.
The ARC5 antenna configuration has five transmitters and two receivers. The phase shift and attenuation of the signal broadcast by each transmitter is measured between the two receivers for a total of five raw phase shifts and five raw attenuations. Because the transmitters are not arranged symmetrically above and below the receivers, conventional borehole compensation cannot be performed. Instead, the ARC5 relies on linear combinations of three transmitters to provide "mixed borehole compensation."  This process results in five calibrated phase-shift and attenuation-resistivity logs which are characterized by the antenna spacings: 10, 16, 22, 28, and 34 in. The 28-in. spacing yields a log identical to a CDR tool.
Vertical resolution is related to the 6-in. receiver spacing that is common to all the measurements, but the depth of investigation increases as the transmitter spacing increases. The result is five phase-shift resistivity logs with different depths of investigation, and five deeper attenuation-resistivity logs, also with different depths of investigation. The phase-shift logs are matched in vertical resolution in conductive beds, but not in resistive beds. Similarly, the attenuation logs are better matched in conductive beds than resistivity beds. This difference in apparent vertical resolution is shown in Fig. 3B.32.
The ARC5 makes a set of measurements at 2 MHz. Larger versions of the tool (6 3/4 and 8 1/4 in.) make measurements at both 2 MHz and 400 kHz. The 400-kHz measurement provides higher signal level in conductive formations (less than 1 ohm•m) and is less sensitive to borehole conditions, particularly in formations where Rt < 1 ohm•m and with OBM. It also has a deeper depth of investigation than the 2-MHz measurement in conductive formations. However, 400-kHz measurements have less sensitivity to Rt in resistive formations than 2-MHz measurements.
Fig. 3B.34 compares ARC5 phase shift and wireline AIT logs. The vertical resolution of both sets of logs is similar, with the deeper AIT 90-in. curve giving a slightly higher value for Rt in the resistive beds. Fig. 3B.35 shows a comparison with the ARC and the ARI dual laterolog in very nonconductive mud. In the bottom track, the ARC logs are inverted for invasion parameters.
In 1989, Teleco introduced the 2-MHz Dual Propagation Resistivity (DPR) tool. This tool measured the phase shift and attenuation at receivers located 27 and 35 in. from a single transmitter (borehole compensation was not used). Teleco was taken over by Baker Hughes, and in 1993 the DPR tool was replaced by the multiple propagation-resistivity (MPR) tool. The MPR configuration consists of upper and lower long- and short-spaced transmitters surrounding a central receiver pair. Antenna spacings range from 23 to 35 in. The two receivers measure the phase shift and attenuation of 2-MHz and 400-kHz signals broadcast by each transmitter. Borehole compensation is performed by averaging the measurements from the symmetrically opposed long and short transmitter pairs. This yields a total of eight logs (long-spaced and short-spaced phase shift and attenuation at 2 MHz and 400 kHz). Resistivity is calculated and displayed either as "apparent" or "borehole-corrected" (hole size and mud-resistivity corrected). Further processing of the logs is available in the MPRteq processing. This processing corrects for a variety of environmental effects. Fig. 3B.36 compares the processed logs with unprocessed logs.
Geosteering With LWD Propagation Tools. Because 2-MHz resistivity measurements are made behind the bit and can be sent uphole in real time, they are often used to steer the drilling of horizontal wells. Before drilling a horizontal well, potential hydrocarbon-bearing zones are first identified using vertical exploration wells. Then the horizontal well is drilled toward a target bed, with marker beds used to maintain the wellbore trajectory. 2-MHz resistivity logs recorded behind the bit are compared to logs from the exploration wells to identify the marker beds. Computer modeling of predicted resistivity-tool response at different well deviation angles is used to modify the well path. This process is called geosteering.
When comparing resistivity logs in a horizontal well to logs from a vertical exploration well in the same zone, the resistivity value often differs in shales and in laminated formations. This difference is caused by anisotropy (the variation of resistivity with direction). In addition to particle-size anisotropy, formations consisting of a series of thin beds of different lithology (such as sequences of sand and shales) also behave anisotropically if a logging tool is significantly longer than the bed thickness. In vertical wells, resistivity tools (conventional induction, 2-MHz and laterologs) read the effective horizontal resistivity, Rh, which can be calculated from the volume average of the layer conductivities (inverse resistivities),
where conductivities are expressed in mS/m. Vsand and Vshale are the bulk volume fractions (percentages) distributed throughout the layered region (layers are all assumed to be approximately uniform in thickness). The effective vertical resistivity, Rv, can be calculated in a similar manner from the volume average of the layer resistivities,
In deviated wells, the apparent resistivity Ra in anisotropic media can be calculated using the approximation
where α is the angle between the tool axis and vertical,
For α = 90° (horizontal wells), Ra = R. For α = 0° (vertical wells), Ra = Rh. Thus, the vertical resistivity cannot be detected at all by conventional resistivity logging tools in vertical wells.
Fig. 3B.37 shows a modeled tool response illustrating differences caused by anisotropy between CDR logs in a vertical well (0° dip) and in a highly deviated well (80° dip). At 0° dip, the CDR log reads Rh. At 80°, the two CDR curves increase in the direction of Rv in the anisotropic bed, with the phase-shift resistivity reading higher than the attenuation resistivity.
Induction and 2-MHz tools both generate azimuthally polarized electric fields that induce current loops that are tilted with respect to the bedding anisotropy. These tilted current loops sense a weighted average of Rv and Rh that depends on dip angle. Extensive modeling and analysis of 2-MHz tool response has demonstrated that radiation effects control the phase-shift measurement more strongly than the attenuation measurement. Thus, separation between 2-MHz phase-shift and attenuation logs provides a good indication of anisotropy (in the absence of invasion and shoulder-bed effect). There is sufficient sensitivity to invert for Rh and Rv only at considerable distances from bed boundaries. Also, note the polarization horn that occurs near the bed boundary at 80°. Polarization horns are a common occurrence at high dip angles, and are an indication in geosteering that the well has crossed into a target bed.
Induction vs. Laterolog Measurements
Laterolog and induction logging tools each have unique characteristics that favor their use in specific situations and applications.
The induction log is generally recommended for holes drilled with only moderately conductive drilling muds or nonconductive muds (e.g., OBM) and for empty or air-drilled holes. The laterolog is generally recommended for holes drilled with very conductive drilling muds (i.e., salt muds).
Induction tools are conductivity-sensitive devices, which are most accurate in low- to medium-resistivity formations. Laterolog tools are resistivity devices, which are most accurate in medium- to high-resistivity formations. In practice, both modern laterolog and induction-logging tools are suitable for most logging conditions, and it is no longer practical to make a specific recommendation for one type in preference to the other, except in extreme conditions.
Laterolog devices see the more resistive zones, and induction tools see the more conductive zones. Therefore, when Rxo is greater than Rt, an induction tool is preferred for Rt determination because laterolog tools will be affected mostly by Rxo. Conversely, a laterolog tool is preferred when Rxo is less than Rt. Conductivity in the borehole has a strong influence on an induction measurement, but little influence on a laterolog measurement.
Starting with the Phasor Induction tool, borehole-corrected logs for induction tools have been available at the wellsite. A caliper and estimate of mud resistivity is essential for induction-borehole correction, either by hand using a chart or automatically.
The AITs have only automatic borehole correction—no charts exist. An analytic forward model was used to compute thousands of cases for the AIT covering the range of each of these parameters. At the wellsite, a caliper and accurate measurement of Rm are used as inputs, and the other two parameters are solved for in a least-squares inversion through the computed table. This method is essential to produce an accurate 10-in. log over a wide range of borehole sizes and mud resistivities.
The following are guidelines for running induction logs, especially array-induction tools:
- A caliper is required in the same toolstring as the induction tool.
- Rm must be measured adequately, preferably downhole, using an accurate sensor. There can be large errors in values of mud resistivity based on surface measurements.
- Adequate standoff is essential. Never run slick.
If these guidelines are followed, modern AITs can give accurate estimates of Rt even when Rxo/Rt is as low as 0.2.
The HRLA array laterolog tool with its inversion of the five array logs has extended the usability of laterolog tools further into the Rxo > Rt region. The AITs, again with inversion of the logs, have extended the induction range in the Rxo < Rt region. Fig. 3B.38 shows the range of usability of the AIT and HRLA tools. In the broad overlap region, both tools can be used. In this region, the HRLA array laterolog tool can be combined with the AIT to determine anisotropic resistivity— Rv and Rh in vertical wells.
When looking at both induction and laterolog logs from the same well or the same field, do not expect the logs to overlay. The Rt values for both tools should be close, but the logs themselves, uncorrected for environmental effects, can be quite different.
When working with older logs, one must keep in mind that both laterolog and induction measurements are influenced by the borehole and by surrounding beds. Surprisingly, thick beds may have some effect on their measurements, depending on shoulder-bed contrast. The measurements of both devices should always be corrected for borehole and surrounding bed effects. Although these corrections are in many cases small, it is good practice to make them routinely. This will ensure that they are not overlooked in the larger number of cases where they are significant.
With either laterolog or induction deep-resistivity measurements, it is essential to record at least three resistivity-log curves with different depths of investigation. With fewer than three competent measurements, it is not possible to make an estimate of the invasion parameters, and Rt and Rxo become guesses. Array-induction and array-laterolog tools make a sufficient number of measurements to use the more rigorous inversion solutions, deriving even more reliable values of Rt and Rxo.
Microresistivity devices measure the resistivity of the flushed zone and delineate permeable beds by detecting the presence of mudcake.
When invasion is moderate to deep, knowledge of Rxo is required to derive Rt from the deep-resistivity measurement. To evaluate a formation with logs, the Rxo/Rt ratio is required for some saturation-estimation methods. In clean formations, a value of the formation resistivity factor F can be computed from Rxo and Rmf if Sxo is known or can be estimated.
Tools designed to measure Rxo have a very shallow depth of investigation, because the flushed zone may extend only a few inches beyond the borehole wall. To avoid the effect of the borehole, a sidewall-pad tool is used. The pad, carrying an array of closely spaced electrodes, is pressed against the formation to minimize the short-circuiting effect of the mud. Currents from the electrodes on the pad must pass through any mudcake to reach the flushed zone.
Microresistivity readings are affected by mudcake; the effect depends on the mudcake resistivity and thickness (hmc). Mudcakes are usually anisotropic, with the resistivity parallel to the borehole wall lower than the resistivity across the mudcake. This increases the mudcake effect on microresistivity readings to make the effective, or electrical, mudcake thickness greater than the physical thickness indicated by the caliper.
Microresistivity measurements have evolved from the first microlog, through the obsolete microlaterolog and proximity-log devices, to the current MicroSFL and Platform Express MCFL microresistivity measurements.
The microlog is still used qualitatively for its ability to detect permeable intervals with a fine vertical resolution, but not for the evaluation of Rxo. The measurement comprises two short-spaced devices with different depths of investigation, providing resistivity measurements of small volumes of mudcake and formation adjacent to the borehole. The presence of a mudcake, identified by a "positive" separation of the two curves, indicates an invaded and, therefore, permeable formation. As a qualitative log, the microlog is usually presented on a linear grid.
Principles. The flexible oil-filled microlog pad is pressed against the borehole wall by arms and springs. The face of the pad has three small in-line electrodes spaced 1 in. [2.5 cm] apart. The electrodes record a 1×1-in. "microinverse" log and a 2-in. "micronormal" log simultaneously.
In an invaded permeable zone, Rmc is usually significantly lower than Rxo. The 2-in. micronormal device has a greater depth of investigation than the microinverse. It is, therefore, less influenced by the mudcake and reads a higher resistivity when mudcake is present. In impermeable formations, the two curves read approximately the same resistivity or have a small negative separation, and the resistivities are usually much greater than in permeable formations.
Positive separation occurs in a permeable zone. Although the microlog curves identify permeable formations, quantitative inferences of permeability are not possible.
Under favorable circumstances, Rxo values can be derived from microlog measurements using charts provided by the service companies. Rmc values for this purpose can be measured directly or estimated from other charts, and h mc is obtained from comparing the caliper curve to bit size. The limitations of the method are as follows:
- The ratio Rxo/Rmc must be less than approximately 15 (porosity more than 15%).
- The value of hmc must be no greater than 0.5 in. [1.3 cm].
- Depth of invasion must be greater than 4 in. [10 cm]; otherwise, the microlog readings are affected by Rt.
MicroSFL LogThe MicroSFL (MSFL) device is a pad-mounted spherically focused logging sensor with two distinct advantages over the microlaterolog and proximity tools it replaced. Unlike these earlier Rxo devices, the MSFL tool is combinable with other logging tools, which eliminates the necessity of a separate logging run to obtain Rxo information. The MSFL log also performs better in shallow invaded zones in the presence of mudcake.
Fig. 3B.39 shows the electrode arrangement (right) and current patterns (left) of the MSFL tool. The surveying current flows outward from a central electrode, A0. Bucking currents, passing between electrodes A0 and A1, flow in the mudcake and the formation. The measuring current, I0, is confined to a path directly into the formation, where it quickly spreads and returns to a remote electrode. By forcing the measure current to flow initially directly into the formation, the effect of mudcake resistivity on the tool response is minimized, yet the tool still has a very shallow depth of investigation.
Synthetic microlog curves (microinverse and micronormal) can be computed from MSFL parameters, because the measure current sees mostly flushed zone and the bucking current sees primarily mudcake.
Environmental Corrections. Although the influence of mudcake on the readings is relatively small, MSFL measurements must be corrected for thickness. Mudcake thickness is normally deduced from a comparison of the actual borehole size, as measured with the caliper, to the known bit size.
The Schlumberger MCFL microresistivity measurements made with Platform Express tool strings are different from previous measurements in several respects:
- Electrodes are mounted on a rigid, mostly metal pad that is not deformed by the borehole wall, allowing a more consistent standoff measurement.
- Survey currents are independently focused in planes parallel and perpendicular to the tool axis, reducing the sensitivity to borehole geometry.
- Three measurements are made with different depths of investigation, which allows a more reliable resolution of mudcake and formation properties with independent response equations.
- The microresistivity sensors are interlaced with the density sensors, so both measurements sample the same volume of formation at the same time.
A vertical resolution of the raw measurements better than 1 in. [2.5 cm] is achieved, and Rxo, Rmc, and hmc are solved simultaneously. Two curves can be displayed in a microlog-like presentation. When the two curves are superimposed, they both read Rxo. Any separation indicates pad standoff from the formation, which is usually caused by mudcake and indicates a permeable formation.
Because Rxo, Rmc, and hmc are obtained directly from the Platform Express microresistivity measurements by inversion processing, no mudcake thickness corrections are required. The values of Rxo can be used directly with medium and deep-resistivity measurements (or array-resistivity measurements) to derive Rt.
Resistivity is the one of the most difficult formation parameters to measure accurately because of the complex changes that occur during and after drilling a well and that may still be occurring during logging. The various components of the downhole environment may have strongly contrasting resistivities, some of which cannot be measured directly, and their physical dimensions may not be readily available. Fig. 3B.1 shows an idealized relationship of the main environmental components. The resistivities and dimensions of all these "layers" (mud, mudcake, flushed zone, and zone of transition) influence all deep-reading resistivity measurements. There is no direct measurement of Rt. It must be inferred from the multiple-depth resistivity measurements.
In a permeable formation, mud resistivity is commonly 1 to 3 orders of magnitude lower than the formation resistivity, or, in the case of OBM, it can be much higher. The downhole mud resistivity can be estimated approximately by measuring the resistivity of a surface sample taken just before mud circulation was stopped before logging and adjusting it for the difference in temperature using an appropriate chart or equation. The shape and size of the borehole and the position of the logging tool in the hole have an influence on resistivity measurements that will not be apparent from a single hole-size measurement.
Attempts to measure the resistivity of an artificially formed sample of mudcake is unlikely to represent in-situ mudcake resistivity accurately. Mudcake thickness cannot be directly measured with current logging tools; it can only be estimated with some uncertainty.
The resistivity of the flushed zone can be measured with reasonable precision if the depth of invasion is greater than the depth of investigation of the Rxo logging tool, but the depth of invasion and the resistivity profile and geometry of the zone of transition are difficult to estimate.
Traditional Rt Methods
Evaluation of the uninvaded formation resistivity Rt is sometimes referred to as invasion correction. It is usually performed assuming a simple three-parameter, step-profile invasion model consisting of a flushed zone of uniform resistivity Rxo with a sharp boundary at the diameter of invasion di to resistivity Rt. This is clearly not realistic, but it allows a complex problem to be solved relatively simply, usually with acceptable accuracy, by using a minimum of three resistivity measurements with different depths of investigation.
A shallow microresistivity measurement, such as the MicroSFL log, is corrected for the influence of the mudcake by using the best available estimates of mudcake thickness and resistivity. It is assumed that di is greater than the depth of investigation of the MicroSFL, so that the MicroSFL log reads only the flushed zone and the mudcake-corrected MicroSFL reading is Rxo.
Next, the deep- and medium-resistivity measurements are corrected for environmental effects using the charts for the tool used. These effects are always corrected in the following order: borehole effect, bed thickness, and shoulder effect. Then the invasion parameters (Rxo, Rt, di) can be found for induction tools using tornado charts. For laterolog tools, butterfly charts are used. Some service companies offer charts for Rt and di when Rxo is known from a microresistivity device.
Inversion for Invasion Parameters
The latest array-induction and array-electrode tools all use some form of inversion (rather than charts) to estimate the invasion parameters—invasion radius or diameter, di, Rxo, and Rt. There are two types of inversion used: 1D and 2D. The formation geometry of each is shown in Fig. 3B.9.
Inversion means first building a parametric model of the formation, then estimating from the log values a "first guess" of the parameter values. Then, a modeling code is used to compute the log response to the model. This is compared with the actual logs. The difference between modeled and measured is used to pick a new set of model parameters, and the log response computed. Again, the modeled logs are compared to the actual logs. This continues until the difference reaches some preset minimum, and then the model parameters are output as the invasion parameters.
Induction response is mostly a function only of coil spacings, with a weak response to the large-scale average formation conductivity. If we can assume that each formation layer is more or less uniformly invaded, then an inversion through a 1D radial forward model will give a close estimate of invasion parameters Rxo and Rt. This is most often true when Rxo > Rt.
Laterolog tools are much more affected at the same time by both bed thickness and invasion parameters in that bed. For this reason, when the formation of interest consists of thin (< 30 ft [10 m]) beds, 2D inversion is necessary for accurate estimation of Rxo, Rt, and di. 1D inversion tends to be used in real-time processing, and 2D inversion is used at the computing center. Some commercial 2D-inversion applications allow very sophisticated choices of parametric models, including transition zones and annulus models.
Inversion methods also return "goodness-of-fit" criteria. At the same time, the modeled logs can be compared with the actual logs as a quality control. Inversion methods also allow the parametric model to be selected to better fit the situation at hand. A four-parameter inversion model, with a zone of transition defined from di to the diameter of the limit of invasion at dj, gives reliable answers in a much wider range of conditions than the traditional three-parameter model. It also generates "quality-of-fit" parameters that indicate when the log readings are not consistent with the model.
Annulus formation is a common phenomenon, caused by the sweeping of conductive ions from the formation by the invading borehole fluid. Fluid-flow models predict annulus formation in a wide range of formation/borehole fluid conditions. Annulus formation has been observed with both WBM and OBM. Even with five or six depths of investigation, the information content is not sufficient to solve for the five annulus parameters (Rxo, Rann, Rt, r1, and r2) independently. However, by invoking the constraint of material balance and supplying an estimate of Rw, the annulus problem can be reduced to three parameters: Rxo, r1, and Rt. r2 is tied to r1 through the material-balance constraint, and Rann can be estimated from Rmf, Rxo, and Rw.
Invasion also changes with time, sometimes rapidly. For this reason, the combination of data taken at different times (e.g., wireline and LWD, or LWD time-lapse), must be done with care. One method that has shown promise is to use a consistent parametric model (step or annulus) and assume Rxo and Rt are constant, allowing only for the invasion radius to change.
Before performing an inversion for invasion parameters, be sure the cause of curve separation is actually invasion. Causes of curve separation include tool-response effects such as shoulder effect, not matching the resolution of the curves, and improper borehole correction. Other formation effects that can cause curve separation include dipping beds and drilling-induced fractures.
The introduction of array-resistivity tools has clearly delineated invasion profiles that were not as expected, even after years of logging in a region. In some regions, it was assumed that the formation was uninvaded because the ILd, ILm, and SFL logs all were very close in pay zones. Logs made by an array-induction tool showed an Rxo < Rt profile. The discrepancy exists because the SFL, being a laterolog, can have a depth of investigation as deep as ILd under Rxo < Rt conditions. The ILm was long considered an inferior measurement to ILd because it would lie lower in resistivity than either SFL or ILd. Modeling shows that either an annulus or Rxo < Rt will produce this curve order.
A powerful method for handling the fundamentally underdetermined problem of invasion correction of resistivity logs is iterative forward modeling. Complex formation geometries and invasion profiles can be worked out by building a model of the best estimate of formation parameters from logs, field knowledge, and petrophysical constraints; modeling the resistivity logs; and varying the parameters until a fit is obtained.
Finally, keep in mind that Rt cannot be measured directly, but must be inferred from multiple-depth resistivity measurements.
Resistivity ImagingMany modern resistivity- and microresistivity-logging tools have arrays of sensors that make multiple measurements, enabling the creation of 2D images of formation resistivity. The images represent resistivity variations with the azimuth around the borehole or with distance away from the borehole.
Inspired by the images produced by early acoustic borehole televiewer tools, borehole resistivity imaging was developed to see actual formation variations, rather than the surface effects that the acoustic images depicted. The first practical resistivity images were produced by an array of closely spaced, shallow-reading button electrodes applied to the borehole wall.
The borehole coverage of this high-resolution microresistivity image was increased on later tools, and deeper reading tools with imaging capabilities have been subsequently developed.
Microresistivity Images. The first microelectrical imaging tool had an array of button electrodes mounted on an enlarged section of the tool mandrel. Electrodes were then placed on two of the four pads of the Dual Dipmeter tool. The Fullbore Formation MicroImager (FMI) tool has button electrodes on all four pads, with extension flaps on each pad to increase the borehole coverage.
FMI image data have sufficient resolution and character to allow using selected electrode signals for conventional dipmeter computations, and the tool’s borehole coverage provides a detailed visual appreciation of geologic features, Fig. 3B.40. Formation dips and fracture orientations can also be derived directly from the images.
Seeing the shape of formation-resistivity variations often provides understanding of the lack of coherent dip found by a dipmeter computation program.
Halliburton’s Electromagnetic MicroImager (EMI) tool is a six-arm resistivity borehole imager tool. Its principle of operation is similar to the FMI tool. Baker Atlas’ Simultaneous Acoustic and Resistivity (STAR) Imager tool integrates resistivity and acoustic borehole imaging sensors into one instrument. The resistivity imager is a six-arm device with powered centralization to keep the acoustic transducer centered. The acoustic sensor also works in OBM where the resistivity imager performance is poor.
ARI Images. The formation resistivity around the borehole is displayed in ARI images as a 2D azimuthal image, with the same dimensions of well depth and azimuthal angle around the well as FMI images. This image has much lower spatial resolution than acoustic or microelectrical images from the UBI and FMI tools, but it complements them well because of its sensitivity to features beyond the borehole wall and its lower sensitivity to shallow features.
AIT Images. The AIT provides images of variations in formation resistivity or conductivity with distance away from the borehole. This capability brings a new dimension to formation image data, because the image contains invasion information useful for understanding how deeply the formation can be invaded.
Radial response functions are used to invert the set of matched vertical resolution logs in the four-parameter invasion model, producing a detailed description of the radial resistivity. Introducing other petrophysical parameters, such as F, Rw, and Rt, and a suitable saturation equation (see Saturation Determination section) allows imaging-computed virgin and invaded-zone saturations.
LWD Resistivity Images
The GVR tool incorporates three 1-in.-diameter azimuthal button electrodes that produce borehole resistivity images during rotary drilling by recording 56 resistivity measurements per rotation with each electrode. The data are processed and recorded downhole for later retrieval.
Because the GVR button electrodes are larger than FMI electrodes and are not in contact with the formation, GVR images are less sharp than FMI images, as seen in Fig. 3B.12. However, often the timeliness of the images more than makes up for the resolution. A compression algorithm allows the images to be sent up in real time for geosteering.
Formation Dip From LWD Images. Dip computation by conventional dipmeter data processing is most effective when the apparent dips (i.e., dips relative to tool inclination) are less than approximately 70°, which is suitable for most vertical and normally deviated wells. LWD has major applications (e.g., geosteering) in highly deviated and horizontal wells, where apparent dips are commonly greater than 70°. A dip-computation process that returns dip from the GVR in real time was developed for high-angle wells. The dip-azimuth and magnitude computations are performed by a robust algorithm in the downhole tool, allowing real-time transmission of the dip data. Although the well deviation is accurately measured by instruments in the drill collar, the relative dip with respect to the bedding is important for geosteering wells along the bedding planes. This method allows improved well placement.
The confidence in computed GVR dips is increased by using data from all three electrodes. Because the electrodes are at fixed distances from each other, irregular tool movement in the hole is unimportant.
The SP curve is a continuous recording vs. depth of the electrical potential difference between a movable electrode in the borehole and a surface electrode. 1 Adjacent to shales, SP readings usually define a straight line known as the shale baseline. Next to permeable formations, the curve departs from the shale baseline; in thick permeable beds, these excursions reach a constant departure from the shale baseline, defining the "sand line." The deflection may be either to the left (negative) or to the right (positive), depending on the relative salinities of the formation water and the mud filtrate. If the formation-water salinity is greater than the mud-filtrate salinity (the more common case), the deflection is to the left.
The relevant features of the SP curve are its shape and the size of its departure from the shale baseline. Because the absolute reading and position of the shale baseline on the log are irrelevant, the SP sensitivity scale and shale-baseline position are selected by the logging engineer for convenience. The SP log is typically scaled at 100 mV per log track. If the resistivities of the mud filtrate and formation water are similar, the SP deflections are small and the curve is rather featureless. An SP curve cannot be recorded in holes filled with nonconductive muds, such as OBMs.
Origin of the SPDeflections of the SP curve are the result of electrochemical and electrokinetic potentials in the formations that cause electric currents to flow in the mud in the borehole.
Electrochemical Component. Membrane Potential. The structure of clay minerals in shales and the concentration of negative electric charges on the clay particle surfaces give shales a selective permeability to electrically charged ions. Most shales act as "cationic membranes" that are permeable to positively charged ions (cations) and impermeable to negative ions (anions).
The upper part of Fig. 3B.41 shows saline formation water in a sandstone formation and mud in the borehole separated by a shale. Sodium chloride, which is usually present in both the formation water and the drilling mud, separates into charged ions (Na+ and Cl–) in solution in water. The Na+ and Cl– ions tend to migrate from a more-concentrated to a less-concentrated solution, but because the intervening shale is a cationic membrane, impervious to Cl– ions, only the Na+ ions can migrate. If, as usual, the formation water is a more concentrated NaCl solution than the mud, there is a net flow of positive ions through the shale from the sandstone to the borehole. This corresponds to a positive electric current in the same direction (indicated by the curved arrow) driven by an electric potential, or electromotive force (EMF), across the shale. Because the shale acts as an ion-selective membrane, the electric potential is known as the membrane potential.
Liquid-Junction Potential. At the edge of the invaded zone, where the mud filtrate and formation water are in direct contact, Na+ and Cl– ions can move freely from one solution to the other. But Cl– ions are smaller and have greater mobility than Na+ ions, so the net diffusion of ions from the more-concentrated formation water to the less-concentrated mud filtrate includes a greater number of Cl– ions than Na+ ions. This is equivalent to a positive current flow in the opposite direction (indicated by the straight arrow at A in Fig. 3B.41.)
The current flowing across the junction between solutions of different salinity is driven by an EMF called the liquid-junction potential. The magnitude of the liquid-junction potential is only approximately one-fifth of the membrane potential.
If the solutions contain substantial amounts of salts other than NaCl, the value of K at 77°F may not be 71. If the permeable formation contains some shale or dispersed clay, the total electrochemical potential, and therefore the SP deflections, is reduced.
Electrokinetic Component. An electrokinetic potential (Ek, also called the streaming potential or electrofiltration potential) is produced when an electrolyte flows through a permeable medium. The size of the electrokinetic potential is determined mainly by the differential pressure producing the flow and the resistivity of the electrolyte.
In the borehole, the electrokinetic potential Ekmc is produced by the flow of mud filtrate through the mudcake deposited on the borehole wall opposite permeable formations. Little or no electrokinetic potential is generated across the permeable formation itself because the differential pressure is usually low. The electrokinetic potential Eksh may, however, be produced across a shale if it has any permeability.
Typically, Ekmc and Eksh are similar in magnitude, and the net electrokinetic contribution to the SP deflection is negligible. If the formation water is fairly saline (resistivity less than 0.1 ohm•m) and the differential pressure is in the normal range of only a few hundred psi, the contribution of the electrokinetic potential can usually be ignored.
Electrokinetic effects may be significant in highly depleted formations or when heavy drilling muds are used because of unusually large differential pressures. Significant electrokinetic effects may also occur in very-low-permeability formations, where an appreciable part of the pressure differential occurs in the formation itself, especially if little or no mudcake is formed. If the formation water is brackish, the mud is resistive, and the low-permeability formation is clean and has some porosity, the electrokinetic effect could be as large as –200 mV.
SP and Permeability
The movement of ions, essential to develop an SP, is possible only in formations with some permeability, however small—a small fraction of a millidarcy is sufficient. There is no direct relationship between the magnitude of the SP deflection and the value of either the formation’s permeability or its porosity.
Static SPThe lower part of Fig. 3B.41 shows SP currents in the borehole and formations. The current directions indicated correspond to the more usual case of formation-water salinity greater than mud-filtrate salinity, producing a potential by the permeable bed lower than the potential by the shale. This corresponds to a deflection to the left on the SP log by the permeable bed.
If the mud-filtrate salinity is greater than the formation-water salinity, the currents flow in the opposite direction, producing positive SP deflections. If the salinities of the mud filtrate and formation water are similar, no SP is generated.
The SP currents flow through four different media: borehole fluid, the invaded zone, the uninvaded part of the permeable formation, and surrounding shales. The SP log measures only the potential drop from the SP currents in the borehole fluid, which may not represent the total SP because there are also potential drops in the formation. If the currents could be interrupted by hypothetical insulating plugs (see the upper part of Fig. 3B.41), the potential observed in the mud would be the total spontaneous potential. This idealized SP deflection is called the static SP (or SSP). The SP deflection practically reaches the SSP in a thick, clean formation.
The borehole presents a much smaller cross-sectional area to current flow than the formations around it, so the resistance of the borehole part of the SP current loop is much higher than the formation part. Nearly all the SP potential drop, therefore, occurs in the borehole if formation resistivities are low-to-moderate and formation beds are thick, so, in practice, the recorded SP deflection approaches the static SP value in thick, permeable beds.
Determination of SSP. To determine the SSP, a sand line is drawn through the maximum (usually egative) excursions of the SP curve adjacent to the thickest permeable beds. A shale baseline is drawn through the SP through the intervening shale beds. The separation of the sand line from the shale baseline, measured in mV, is the SSP. Any SP anomalies are discounted.
If there are no thick, clean, permeable invaded beds in the zone under study, the SP reading can be corrected for the effects of bed thickness and invasion to estimate the SSP by using charts available from service companies.
Shape of the SP Curve. The slope of the SP curve is proportional to the intensity of the SP currents in the borehole at that depth. Because the current intensity is highest at the boundaries of the permeable formation, the slope of the SP curve is at a maximum, and an inflection point occurs at these bed boundaries.
The shape of the SP curve and the amplitude of its deflection in permeable beds depend on the following factors: thickness and true formation resistivity of the permeable bed, resistivity of the flushed zone (Rxo) and diameter di, resistivity of the adjacent shale bed (Rs), and resistivity of the mud and the diameter of the borehole (dh).
Fig 3B.42 shows examples of SP curves computed for Rt = Rs = Rm (on the left) and Rt = Rs = 21Rm (in the center). In the first case (Rt = Rs = Rm), the SP curve gives a much sharper definition of the boundaries of the permeable beds, and the SP deflections approach the SSP value more closely than in the case where the formation-to-mud resistivity ratio is 21.
Highly Resistive Formations. Highly resistive formations between some shales and permeable beds can significantly alter the distribution of the SP currents and change the expected shape of the SP curve. The currents shown flowing from shale bed Sh 1 toward permeable bed P2 in Fig. 3B.43 are largely confined to the borehole by the high resistivity of the formation separating Sh1 and P2. The current in the borehole over this interval is constant, so for a constant borehole diameter, the SP curve is a straight line inclined to the shale baseline.
Shale-Baseline Shifts. A shift of the shale baseline can occur when formation waters of different salinities are separated by a shale bed that is not a perfect cationic membrane. Fig. 3B.44 shows an SP log recorded in a series of sandstones (B, D, F, and H) separated by thin shales or shaly sandstones (C, E, and G). The SSP of Sandstone B is –42 mV. Shale C is not a perfect cationic membrane, and the SP curve does not return to the shale baseline defined by Shale A. A new shale baseline defined by Shale E gives SP deflections of +44 mV in Sandstone D and –23 mV in Sandstone F.
Baseline shifts also occur when formation waters of different salinities are separated by an impermeable layer that is not a shale. In this case, the SP curve shows little or no variation at the level of the change in salinity, but the deflections at the upper and lower shale boundaries are different and may even have different polarities.
Invasion-Related Anomalies. If the mud filtrate and the formation water have significantly different salinities, and therefore different densities, gravity-induced fluid migration can cause SP anomalies in highly permeable formations, as shown in Fig. 3B.44. Invasion is very shallow near the lower boundary of each permeable interval and deeper near the upper boundary.
The SP curve is rounded at the upper boundary because of the deep invasion, and it may have a sawtooth profile at thin, impervious shale streaks in which the SP deflection exceeds the SSP above the shale streak. A reading greater than the SSP is caused by the accumulation of filtrate below the shale streak. Encircling the hole is a horizontal disk of shale sandwiched between salt water and fresher mud filtrate that acts like a battery cell. The EMF of this cell is superimposed on the normal SSP, producing the sawtooth profile.
Noisy SP Logs. SP measuring circuits are sensitive and therefore prone to recording spurious electrical noise superimposed on the SP curve. Occasionally, the source of noise cannot be eliminated during logging, and a noisy log is recorded. However, this does not always render the log unusable.
A regular sine-wave signal may be superimposed on the SP curve when some part of the logging winch is magnetized. An intermittent contact between the casing and cable armor may also cause spurious spikes on the SP curve. In these situations, the SP curve can usually be read so that the sine-wave amplitude or noise spikes are not added to or subtracted from the authentic SP deflection.
Direct currents flowing through formations near the SP electrode can cause erroneous SP readings, particularly where formation resistivities are high. These currents may be caused by "bimetallism," when the logging tool has exposed metal housings. The currents are small and have a significant effect on the SP only in highly resistive formations. If an SP curve looks questionable in highly resistive formations, it should be relied on only in lower-resistivity intervals.
The offshore logging environment is notorious for its ample supply of sources of electrical noise, such as wave motion, cathodic protection systems, rig welding, onboard generators, and leaky power sources. On land, proximity to power lines and pumping wells may have a similar effect on the SP curve, but the effects can usually be minimized by carefully choosing the ground-electrode location.
Uses and Interpretation of Well Logs
Determination of SaturationWater saturation is the fraction of the pore volume of the reservoir rock that is filled with water. It is generally assumed, unless otherwise known, that the pore volume not filled with water is filled with hydrocarbons. Determining water and hydrocarbon saturation is one of the basic objectives of well logging.
Clean Formations. All water saturation determinations from resistivity logs in clean (nonshaly) formations with homogeneous intergranular porosity are based on Archie’s water saturation equation, or variations thereof. The equation is
Rw is the formation water resistivity, Rt is the true formation resistivity, and F is the formation resistivity factor. F is usually obtained from the measured porosity of the formation through the relationship
For Sxo, the water saturation in the flushed zone, a similar expression exists:
where Rmf is the mud filtrate resistivity and Rxo is the flushed zone resistivity.
For simplicity, the saturation exponent n is usually taken as 2. Laboratory experiments have shown that this is a reasonable value for average cases. For more exacting work, electrical measurements on cores will produce better numbers for n, a, and m. When core measured values are unavailable, the values of a and m in Eq. 3B.7 can be estimated as follows: in carbonates, F = 1 / ϕ2 is usually used; in sands, F = 0.62 / ϕ2 (Humble formula), or F = 0.81 / ϕ2 (a simpler form practically equivalent to the Humble formula). These equations are easily programmed into spreadsheets and are available in most log interpretation software.
The accuracy of the Archie equation, Eq. 3B.10 and its derivatives, depends in large measure, of course, on the accuracy of the fundamental input parameters: Rw, F, and Rt. The deep resistivity measurement (induction or laterolog) must be corrected, therefore, for borehole, bed thickness, and invasion (see Sections 3B.8 and 3B.9). It is almost never safe to make the assumption "deep = Rt." The most appropriate porosity log (sonic, neutron, density, magnetic resonance, or other) or combination of porosity and lithology measurements must be used to obtain porosity, and the proper porosity-to-formation factor relationship must be used. Finally, the Rw value should be verified in as many ways as possible: calculation from the SP curve, water catalog, calculation from nearby water-bearing formation, and/or water sample measurement.
Alternate methods for determining water saturation include analysis of cores cut with low-invasion OBMs and single-well chemical-tracer tests (described in the chapter on single-well chemical-tracer testing in this section of the Handbook). These independent methods can be used to calibrate log analyses.
Resistivity-vs.-Porosity Crossplots. Combining Eqs. 3B.10 and 3B.11 , the Archie saturation equation may be written
If n and m are equal to 2, and a = 1, then
Eq. 3B.14 shows that for Rw constant, ϕSw is proportional to is the quantity of water per unit volume of formation. To emphasize the proportionality between ϕ and , Eq. 3B.14 may be rewritten:
For a 100% water-saturated formation, Sw = 1 and Rt = R0. If R0 for water-saturated formations is plotted on an inverse square-root scale vs. ϕ, all points should fall on a straight line given by .
Furthermore, the points corresponding to any other constant value of Sw will also fall on a straight line, because in Eq. 3B.14 the coefficient is constant for constant values of Rw and Sw.
Fig. 3B.45 shows several points plotted over an interval in which formation-water resistivity is constant (as indicated by constant SP deflections opposite the thick, clean permeable beds). Assuming that at least some of the points are from 100% water-bearing formations, the line for Sw = 1 is drawn from the pivot point (ϕ = 0, Rt = ∞) through the most northwesterly plotted points. The slope of this line defines the value of Rw as shown on Fig. 3B.45, for ϕ = 10%, R0 = 6.5 ohm•m. For this formation, the most appropriate F – ϕ relation is F = 1/ϕ2. Thus, for ϕ = 10%, F = 100. Because Rw = R0/F, Rw = 0.065 ohm•m, as shown.
If the matrix composition remains constant over the formations under investigation, the basic measurement from the sonic, density, or neutron logs can be plotted directly vs. Rt with similar results. This is possible because of the linear relationship between porosity and bulk density, sonic transit time, or neutron-hydrogen index response. An example of a sonic-induction crossplot is shown in Fig. 3B.46. The transit time has been plotted against the induction resistivity for several levels. The northwesterly points define the 100% water saturation line. The transit-time value at the point where this line intersects the horizontal line of infinite resistivity is the matrix-transit time, tma In Fig. 3B.46, tma is found to be approximately 47.5 μs/ft (156 μs/m). This corresponds to a matrix velocity of 21,000 ft/sec (6,400 m/s).
By knowing tma, a porosity scale, a scale of formation factor (e.g., from F = 1/ϕ2) can be easily derived. A vertical line drawn through F = 100 (or ϕ = 10) intersects the water line at R0 = 5 ohm•m; accordingly, Rw (= R0/F) is 0.05 ohm•m.
The lines for other Sw values are straight lines, determined as previously described, radiating out from the Rt =∞, tma = 47.5 pivot point.
Density and neutron logs can be crossplotted against resistivity in a manner identical to the sonic logs. For density logs, the intersection of the 100% water line with the infinite-resistivity line yields the matrix-density value, ρma. For neutron logs, the intersection defines the matrix-hydrogen index, or apparent matrix porosity. Knowledge of matrix density or hydrogen index permits the ρB or ϕN scale to be rescaled in ϕ and F units. With the F scale defined, Rw can be calculated as for the sonic-resistivity crossplot, and lines of constant water saturation can be constructed in a similar manner.
These resistivity-vs.-porosity crossplots require that formation water resistivity be constant over the interval plotted, that lithology be constant, that invasion not be deep, and that the measured-porosity log parameter (i.e., t, ρB, or ϕN) can be linearly related to porosity. This last condition implies that the time-average transform for the conversion of t into porosity is appropriate.
The neutron-resistivity crossplot is not as satisfactory in gas-bearing formations as are the sonic- or density-resistivity crossplots. The apparent porosity measured by the neutron log in gas zones is often much too low. This results in overstated Sw values in gas zones. Indeed, in a gas zone, the neutron resistivity may indicate a porous gas-bearing zone to be near zero porosity and 100% water bearing. In contrast, the sonic- or density-resistivity tends to be slightly optimistic in gas zones (i.e., porosities may be slightly high and water saturations slightly low).
Microresistivity-vs.-Porosity Crossplots. This method is particularly useful for older logs or cases in which the analyst has only a paper copy of the log. A resistivity-porosity plot can also be made using the values from a shallow-investigation resistivity log such as the microlaterolog, MSFL, or MCFL log. If the microresistivity log reads approximately Rxo, then a line through points of mud-filtrate-saturated formations (Sxo = 1) should have a slope related to Rmf. Rmf is an important parameter, and this check of its value by means of a sonic-microresistivity or density-microresistivity crossplot is often useful.
These plots are also valuable for improved determinations of matrix parameters (either tma or ρma), particularly in cases where the sonic-resistivity or density-resistivity plot does not give a clear answer because of hydrocarbon saturation. The F Rmf line should be easier to determine because Sxo is usually fairly high even in hydrocarbon-bearing formations.
Fig. 3B.47 shows a resistivity-porosity plot in which both the deep induction reading and the microlaterolog at the same levels are plotted in a series of water-bearing formations. The porosity values were derived in this case from a neutron-density crossplot. The plots from the two logs define two trends corresponding respectively to Sw = 1 (using deep induction) and Sxo = 1 (using microlaterolog data). The points not in these trends can be divided into two groups:
- Points whose microlaterolog readings fall on the Sxo = 1 line but whose deep induction log readings fall below the Sw = 1 line (Points 2, 9, and 10) are probably the result of either deep invasion or adjacent-bed effect in which deep resistivity is greater than Rt.
- Points whose induction log readings fall on the Sw = 1 line but whose microlaterolog points fall above the Sxo = 1 line are possibly a result of shallow invasion in which RMLL is lower than Rxo.
Resistivity-porosity plots are thus often more informative if the short-spaced resistivity or medium-induction values are also plotted. Not only does this permit an appreciation of invasion effects, but it may also indicate moved oil.
Rwa ComparisonIf water saturation is assumed to be 100%, the Archie water saturation equation (Eq. 3B.10) reduces to
The term Rwa is used in Eq. 3B.16 rather than Rw to indicate that this is an apparent formation water resistivity. It is only equal to Rw in 100% water-bearing formations. In hydrocarbon-bearing formations, Rwa computed from Eq. 3B.12 will be greater than Rw. Indeed, by combining Eqs. 3B.16 and 3B.10, the relationship between Sw, Rwa, and, Rw can be shown to be
The Rwa technique can, therefore, be useful for identifying potential hydrocarbon-bearing zones and for obtaining Rw values.
In practice, Rwa is obtained by simply dividing the deep induction resistivity (or deep laterolog resistivity) by the formation factor obtained from a porosity log or a combination of porosity logs. Today, a continuous Rwa computation is made over a long interval of the borehole in real time. If one has only paper logs, many individual manual computations are made so as to approximate a continuous computation.
Resistivity-Ratio Methods. In resistivity-ratio methods, it is assumed that a formation is divided into two distinct regions—a flushed zone and a noninvaded zone. Both zones have the same F, but each contains water of a distinct resistivity (Rmf in the invaded zone and Rw in the noninvaded zone). The resistivities of the two zones must be measurable or derivable from logs, and methods for determining the resistivity of the water in each zone must be available.
Because of the necessary assumptions, the resistivity-ratio methods have limitations, but when no porosity or formation factor data are available, they are sometimes the only choice. The principal limitation arises from the inability of any resistivity device to measure either Rx or R, totally independent of the other. Simply put, invasion must be deep enough to allow a shallow investigating resistivity device to measure Rxo but not so deep that a deep-resistivity device cannot measure Rt.
Another difficulty appears when hydrocarbons are present. In this case, some knowledge or assumption of the value of the flushed or invaded zone saturation is necessary.
Flushed-Zone Method. If n = 2 is assumed and Eq. 3B.10 is divided by Eq. 3B.12,
This equation gives the ratio of Sw to Sxo, and no knowledge of formation factor or porosity is needed. Rxo may be found from a microresistivity log, Rt from an induction or laterolog, and Rmf/Rw from measured values or from the SP curve.
The ratio is valuable in itself as an index of oil movability. If Sw/Sxo = 1, then no hydrocarbons have been moved by invasion, whether or not the formation contains hydrocarbons. If Sw/Sxo is approximately 0.7 or less, movable hydrocarbons are indicated. The value of Sw/Sxo, along with ϕ and Swo, is useful in evaluating reservoirs.
To determine Sw from Eq. 3B.18, Sxo must be known. For moderate invasion and average residual oil saturation, an empirical relation between Sw and Sxo has been found useful: Sxo = Sw1/5. Inserting this into Eq. 3B.18 gives:
Service companies provide charts for graphical solution of this equation, or it can be easily programmed into a spreadsheet.
Invaded-Zone Method. The invaded-zone method is useful for water saturation determination when only an ES, IES, or other early-resistivity log is available and no porosity-log or formation-factor data exist. (This section also uses some early nomenclature.) For application of the method, Ri/Rm must be at least 10.
Archie’s equation for the invaded zone is
where Rz is the resistivity of the water in the invaded zone. Because of incomplete flushing, Rz is a mixture of mud filtrate, Rmf, and formation water, Rw.
Studies of many logs suggest that Si and Sw are related by
Dividing the noninvaded-zone water saturation equation (Eq. 3B.10) by Eq. 3B.20 and using the relationship presented in Eq. 3B.21 yields an expression for Sw:
To use Eq. 3B.22, Rt is taken from a deep resistivity device such as a deep induction or deep laterolog (corrected for borehole effect and bed thickness). Ri is taken from a shallow resistivity device such as a Laterolog 8, 16-in. normal, or SFL (corrected for borehole effect and bed thickness).
Rz is given by the relationship
where z is the fraction of the invaded zone pore water, which is formation water, and 1 – z is the fraction that is mud filtrate. Experience has indicated that z varies from 0.075 in cases of normal invasion to 0.035 in cases of deep invasion or vuggy formations.
Fig. 3B.48 solves Eq. 3B.22. It is entered with Rmf/Rw on the appropriate z scale and Ri/Rt (oblique lines) to determine Sw. When Ri/Rt is close to unity, some caution is required. The formation may be extremely invaded or there may be little invasion, or it may be dense and impermeable. On the other hand, many good hydrocarbon-bearing reservoirs will have Ri/Rt ≈ 1.
Rxo/Rt QuicklookThe Rxo/Rt quicklook method can be used to identify hydrocarbon-bearing formations and to indicate hydrocarbon movability (producibility). When Sw/Sxo is 1 in a permeable zone, the zone will produce water or be nonproductive regardless of water saturation. A value Sw/Sxo significantly less than 1 indicates that the zone is permeable and contains some hydrocarbons, and that the hydrocarbons have been flushed (moved) by invasion. Thus, the zone contains producible hydrocarbon.
Eq. 3B.18 can be written as
which shows that an indication of Sw/Sxo can be obtained by comparing Rxo/Rt with Rmf/Rw, where the subscript SP emphasizes that Rmf/Rw is derivable from the SP. Equivalently, the comparison can be between log Rxo/Rt and the SP curve for an indication of log Sw/Sxo.
The value of log Rxo/Rt is computed from solving the three or more resistivity logs for invasion parameters. It is used as an overlay comparison curve with the SP. Separations between the log Rxo/Rt curve, properly scaled to match the SP, and the SP curve provide a quick-look location of producible hydrocarbons.
Originally, log Rxo/Rt was computed from RLL8/RID or RSFL/RID. Use was made of the observation that over a wide range of invasion diameters (from approximately 20 to 100 in.), Rxo/Rt depends primarily on the value of RLL8/RID or RSFL/RID. The relationship used for the LL8 device was
For the SFL device, it was
Much more sophisticated algorithms are now used to obtain Rxo/Rt. These values are output in real time as separate logs.
To interpret the Rxo/Rt quick-look curve, the impermeable zones must be eliminated by reference to the SP, GR, or microlog curves or by resistivity ratios. Then, if the SP and Rxo/Rt (actually –K log Rxo/Rt) curves coincide in a permeable zone, the zone will most probably produce water. If, however, the Rxo/Rt curve reads appreciably lower (i.e., to the right) than the SP, the zone should produce hydrocarbons. An Rxo/Rt value less than the SP amplitude indicates movable hydrocarbons are present.
The Rxo/Rt quick-look technique is applicable to fresh mud conditions (Rxo > Rt) in formations where invasion falls within the limits demanded by the Rxo/Rt computation. For the simpler computation technique using Eq. 3B.25 and –25, that is for di 30 to 70 in.; for the more sophisticated techniques, that is, between 20 and 120 in. Even in the more restrictive case, however, any errors are optimistic. In other words, water zones may appear to be hydrocarbon-productive. This constitutes a safeguard against overlooking pay zones, and it is considered a desirable feature in any quick-look approach.
The Rxo/Rt technique efficiently handles variations in formation water resistivity, Rw, and in shaliness. Any change in Rw is reflected similarly into both the computed Rxo/Rt and the SP amplitude. Thus, comparing the two curves still permits formation-fluid identification. Shaliness also affects the two curves in a similar manner. All other things remaining constant, shaliness reduces the Rxo/Rt value and the SP amplitude. Finally, the Rxo/Rt quick-look technique does not require porosity data, nor use of any F – ϕ relationships.
Fig. 3B.49 is an example of a shaly gas sand at 3,760 through 3,788 ft and several water-productive sands with varying amounts of shaliness. The productive-gas sand is identified by the separation between the Rxo/Rt and SP curves. Water-productive zones are shown by lack of separation. In shaly water zones, the variation in the SP curve is essentially the same as the variation in the Rxo/Rt ratio—a result of the same shale. Therefore, the comparison is not significantly disturbed by shaliness. Neither is it disturbed by variations in Rw.
Estimates of water saturation and saturation ratio in clean formations can be made by comparing the Rxo/Rt and SP curves. Eq. 3B.24 permits Sw/Sxo to be estimated, and then Eq. 3B.19 allows Sw to be estimated.
Shaly-sand analysis has been the subject of much continuing work over the past 20 years, and a detailed coverage of that work is far beyond the scope of this Handbook. We will present the problem to be solved and a couple of methods that can be applied to older logs where detailed information is not available.
Shales are one of the more important common constituents of rocks in log analysis. Aside from their effects on porosity and permeability, this importance stems from their electrical properties, which have a great influence on the determination of fluid saturations.
Shales are loose, plastic, fine-grained mixtures of clay-sized particles or colloidal-sized particles and often contain a high proportion of clay minerals. Most clay minerals are structured in sheets of alumina-octahedron and silica-tetrahedron lattices. There is usually an excess of negative electrical charges within the clay sheets. The substitution of Al+++ by ions of lower valence is the most common cause of this excess; the structure of the crystal remains the same. This local electrical imbalance must be compensated to maintain the electrical neutrality of the clay particle. The compensating agents are positive ions—cations or counterions—which cling to the surface of the clay sheets in a hypothetical dry state. The positive surface charge is usually measured in terms of milli-ions equivalents per 100 grams of dry clay minerals and is called the cation exchange capacity (CEC). When the clay particles are immersed in water, the Coulomb forces holding the positive surface ions are reduced by the dielectric properties of water. The counterions leave the clay surface and move relatively freely in a layer of water close to the surface (the electrical balance must be maintained so that the counterions remain close to the clay water interface) and contribute to the conductivity of the rock.
The Archie water saturation equation, which relates rock resistivity to water saturation, assumes that the formation water is the only electrically conductive material in the formation. The presence of another conductive material (i.e., shale) requires either that the Archie equation be modified to accommodate the existence of another conductive material, or that a new model be developed to relate rock resistivity to water saturation in shaly formations. The presence of clay also complicates the definition or concept of rock porosity. The layer of closely bound surface water on the clay particle can represent a very significant amount of porosity. However, this porosity is not available as a potential reservoir for hydrocarbons. Thus, a shale or shaly formation may exhibit a high total porosity, yet a low effective porosity as a potential hydrocarbon reservoir.
The way shaliness affects a log reading depends on the amount of shale and its physical properties. It may also depend on the way the shale is distributed in the formation. Shaly material can be distributed in the formation in three ways:
- Shale can exist in the form of laminae between which are layers of sand. The laminar shale does not affect the porosity or permeability of the sand streaks themselves. However, when the amount of laminar shale is increased and the amount of porous medium is correspondingly decreased, overall average effective porosity is reduced in proportion.
- Shale can exist as grains or nodules in the formation matrix. This matrix shale is termed structural shale; it is usually considered to have properties similar to those of laminar shale and nearby massive shales.
- The shaly material can be dispersed throughout the sand, partially filling the intergranular interstices. The dispersed shale may be in accumulations adhering to or coating the sand grains, or it may partially fill the smaller pore channels. Dispersed shale in the pores markedly reduces the permeability of the formation.
All these forms of shale can, of course, occur simultaneously in the same formation.
Over the years, a large number of models relating resistivity and fluid saturations have been proposed. Many have been developed assuming the shale exists in one of the three specific geometric forms. All these models are composed of a clean sand term, described by the Archie water saturation equation, plus a shale term. The shale term may be fairly simple or quite complex; the shale term may be relatively independent of, or it may interact with, the clean sand term. All the models reduce to the Archie water saturation equation when the fraction of shale is zero; for relatively small amounts of shaliness, most models and methods yield quite similar results.
Only a very few of these models will be reviewed here to provide some flavor and understanding for the evolution of shaly-sand interpretation logic.
Laminated Sand/Shale – Simplified ModelIn this laminar shale model, Rt, the resistivity in the direction of the bedding planes, is related to Rsh (the resistivity of the shale laminae) and to Rsd (the resistivity of the clean sand laminae) by a parallel resistivity relationship,
where Vlam is the bulk-volume fraction of the shale, distributed in laminae, each of more-or-less uniform thickness.
For clean-sand laminae, , where Fsd is the formation resistivity factor of the clean sand. Because (where ϕsd is the sand-streak porosity) and f = (1– Vlam )ϕsd (where ϕ is the bulk-formation porosity), then
To evaluate Sw by the laminated model, Rt, Rw, ϕ, Vlam, and Rsh must be determined.
For the determination of Rt, the problem is the same as for clean formations. If Rw is not known, its determination usually involves looking at a nearby clean sand and solving for Rw using the SP measurement. If the formation is water-bearing, the resistivity and porosity measurements can be used.
For the determination of ϕ and Vlam, a combination of porosity logs can be used. For example, as illustrated in Fig. 3B.50, a crossplot of ϕN and ϕB from a density log is effective. The triangle of the figure is defined by the matrix point, water point, and shale point. In this example, the matrix point is at ϕN = 0 (the neutron log was scaled in apparent sandstone porosity) and ϕma = 2.65 g/cm3 (quartz matrix). The shale point is at ϕN = 50 p.u. and ϕsh = 2.45 g/cm3. These values were taken in a nearby thick shale bed; it is assumed that shale laminae in the shaly sand under investigation are similar to the nearby massive shale beds. The water point is, of course, located at ϕN = 100 p.u. and ϕB = 1 g/cm3. The matrix-water line and shale-water line are linearly divided into porosity; the matrix-shale line and water-shale line are linearly divided into shale percentages.
Point A, plotted as an example, corresponds to log readings of ϕB = 2.2 g/cm3 and ϕN = 33 p.u. Interpretation by the lines on the plot yields 23% and Vsh (or Vlam) = 16 %.
Direct use of this crossplot assumes 100% water saturation in the zone investigated by the tools. Because oil has a density and hydrogen content normally not greatly different from water, this crossplot technique can be used with acceptable accuracy in oil-bearing formations. The presence of gas or light hydrocarbon, however, decreases ϕN and decreases ϕB. This would cause the point to shift in a northwesterly direction. When gas or light hydrocarbons are present, an additional shaliness indicator, such as GR or SP data, is needed to evaluate the amount of the shift.
Using the laminated model, an equation for Rxo analogous to Eq. 3B.28 could be written. Sxo would replace Sw, and Rmf would replace Rw. The other terms (ϕ, Vlam, and Rsh) remain the same in the two equations. Assuming Sxo = Sw1/5 (as in the flushed-zone ratio method) and the ratio of the PSP (SP deflection in the shaly sand) to the SSP (SP deflection in a nearby clean sand of similar formation water) is a measure of shaliness, Vlam, water saturation could be calculated from Rxo/Rt and PSP in the shaly sand and SSP (or Rmf/Rw) in a nearby clean sand.
In this model, the formation conducts electrical current through a network composed of the pore water and dispersed clay. As suggested by de Witte, it seems acceptable to consider that the water and the dispersed shale conduct an electrical current like a mixture of electrolytes. Development of this assumption yields
where ϕim = intermatrix porosity, which includes all the space occupied by fluids and dispersed shale; Sim = the fraction of the intermatrix porosity occupied by the formation-water, dispersed-shale mixture; q = the fraction of the intermatrix porosity occupied by the dispersed shale; and Rshd = the resistivity of the dispersed shale. Also, it can be shown that Sw = (Sim − q)/(1 − q), where Sw is the water saturation in the fraction of true effective formation porosity.
Combining these relations and solving for Sw yields
Usually, ϕim can be obtained directly from a sonic log because dispersed clay in the rock pores is seen as water by the sonic measurement. The value of q can be obtained from a comparison of a sonic and density log. Indeed, if ρshd ≅ ρma, then qsv ϕ (ϕSV − ϕD)/ ϕSV, where ϕSV and ϕD are the sonic and density derived porosities, respectively. In this case, ϕD approximates ϕ, the effective porosity available for fluid saturation.
The value of Rsh is more difficult to evaluate. It is usually taken as equal to Rsh in nearby shale beds. Fortunately, its value is not too critical if it is at least several times greater than Rw. In fact, when Rw is small compared to Rsh and the sand is not too shaly, Eq. 3B.30 can be simplified to a form independent of Rsh:
Total Shale Relationship
Based upon the previously described ideas, laboratory investigations, and field experience, it has been found that a simple relationship of the following form works well for many shaly formations independent of the distribution of the shale and over the range of Sw values encountered in practice:
In using this equation, Rsh is taken equal to the resistivity of the adjacent shale beds, and Vsh is the shale fraction as determined from a total shale indicator.
Before the Waxman-Smits formulation, equations of the form of Eq. 3B.29 and 3B.32 gained wide acceptance in the evaluation of shaly sands. These equations have a general form of
where α denotes a predominant sand term that is dependent on the amount of sand, its porosity, and the resistivity of the saturating water. The sand term always reduces to Archie’s water saturation equation when the shale fraction is zero. γ denotes a predominant shale term that depends on the amount and resistivity of the shale.
Dual Water Models
In 1968, Waxman and Smits proposed, based on extensive laboratory work and theoretical study, a saturation-resistivity relationship for shaly formations that related the resistivity contribution of the shale (to the overall resistivity of the formation) to the CEC of the shale. The Waxman-Smits relationship is
where F * is the formation factor of the interconnected porosity, Sw also relates to the interconnected pores, B is the equivalent conductance of the sodium clay-exchange cations as a function of the formation water conductivity, and Qv is the CEC of the rock per unit pore volume.
Unfortunately, a continuous in-situ measurement of rock CEC was not available when this study was presented. As a result, the dual water model was developed as a practical solution. The dual water method is based on three premises:
- The conductivity of clay is because of its CEC.
- The CEC of pure clays is proportional to the specific surface area of the clay.
- In saline solutions, the anions are excluded from a layer of water around the surface of the grain. The thickness of this layer expands as the salinity of the solution (below a certain limit) decreases, and the thickness is a function of salinity and temperature.
Therefore, because CEC is proportional to specific area (area per unit weight) and to the volume of water in the counter-ion exclusion layer per unit weight of clay. Consequently, the conductivity of clay is proportional to the volume of the counter-ion exclusion layer, this layer being "bound" to the surface of the clay grains. For clays, this very thin sheet of bound water is important because of the large surface areas of clays relative to sand grains (several magnitudes greater). Therefore, in the dual water model, a clay is modeled as consisting of two components: bound water and clay minerals.
The clay minerals are modeled as being electrically inert; the clay electrical conductivity is modeled as being derived from the conductivity of the bound water, Cwb. Cwb is assumed to be independent of clay type (from the second postulate described previously). The amount of bound water varies according to clay type, being higher for the finer clays (with higher surface areas), such as montmorillonite, and lower for coarser clays, such as kaolinite. Salinity also has an effect; in low-salinity waters (roughly < 20,000 ppm NaCl), the diffuse layer expands.
The bound water is immovable under normal conditions; therefore, the volume it occupies cannot be displaced by hydrocarbons. Because the clay minerals (dry colloids) are considered electrically inert, they may be treated just as other minerals. Schematically, shaly formations are modeled with the dual water model, as illustrated in Table 3B.1.
For most rocks (except for conductive minerals such as pyrite, which cannot be treated in this way) only the porous part needs to be considered when discussing electrical properties, and it is treated according to the Archie water-saturation equation. The equation becomes
where a, m, and n have the usual Archie connotations. σt is the conductivity of the noninvaded, virgin formation (1/Rt), and σwe is the equivalent conductivity of the waters in the pore space.
Note that ϕt and Swt refer to total pore volume; this includes the pore volumes saturated with the bound water and the formation connate water (sometimes called the "free" water). The equivalent water conductivity, σwe, is
where Vw and Vwb are the bulk volumes of formation water and bound water, respectively, and σw and σwb are their conductivities.
In terms of saturation, Eq. 3B.36 becomes
where Swb is the bound water saturation (i.e., the fraction of the total pore volume occupied by the bound water).
Eq. 3B.39 describes the equivalent-water conductivity as a function of the formation water conductivity plus the bound-water conductivity. The saturation equation (Eq. 3B.35) becomes
The porosity and water saturation of the sand (clean formation) phase (that is, the nonclay phase) of the formation is obtained by subtracting the bulk-volume fraction of bound water (ϕt Swb). Therefore, the effective porosity is
and the water saturation is
To evaluate a shaly formation using the dual water model, four parameters must be determined. They are σw (or Rw), σwb (or Rwb), ϕt, and Swb. A neutron-density crossplot provides a good value of ϕt. Swb is obtainable from a variety of shale-sensitive measurements (SP, GR, ϕN, Rt, ϕN – ρB, t – ρB, etc.). Rwb and Rw are usually determined by the log analyst and entered as input parameters.
This chapter presents the fundamentals of the various logging tools used to measure formation resistivities, conductivities, and naturally occurring currents that normally exists in wellbores containing conductive fluids. The borehole and formation environments are described, along with their effect on log response and what can be done to determine true formation resistivity. Simplified manual methods for interpreting log responses are presented, along with a description of the expanded amount of information that can now be generated with sophisticated computer programs. It has been the objective to provide sufficient insight to use existing log data and to indicate what tools are available to capture data in new wells. Other sections in this petrophysics chapter contain information on other types of logs that are needed in conjunction with resistivity logs to obtain an understanding of reservoir rock and fluid properties.
Service companies, oil companies, and many third parties have developed software to calculate Sw, in most situations. Planning logging jobs will always be a balance of the types of tools available, the data needed, and the costs of acquisition. In the long run, it is best to include all tools that provide sufficient data for the formation at hand.
|amf||=||mud filtrate chemical activity|
|aw||=||formation water chemical activity|
|di||=||diameter of invasion (in., m)|
|Ekmc||=||electrokinetic potential of the mudcake|
|Eksh||=||electrokinetic potential of shale|
|F||=||formation factor relating resistivity to porosity|
|G||=||induction integrated radial-response function|
|I||=||electrical current, Amperes|
|Rann||=||resistivity of the annulus|
|Rh||=||resistivity in the horizontal direction (ohm•m)|
|Rm||=||resistivity of the mud column (ohm•m)|
|Rmc||=||resistivity of the mudcake|
|Rmf||=||resistivity of the mud filtrate|
|Rxo||=||resistivity of the invaded zone|
|Rt||=||resistivity of the uninvaded formation|
|Rv||=||resistivity in the vertical direction (ohm•m)|
|Rw||=||resistivity of the formation connate water (ohm•m)|
|Rwa||=||apparent water resistivity from deep resistivity and porosity|
|Sxo||=||water saturation of the invaded zone|
|Sw||=||water saturation in the uninvaded zone|
|t||=||acoustic travel time (μs/ft)|
|tma||=||acoustic travel time of the rock matrix(μs/ft)|
|V||=||electrical voltage, volts|
|Vsd||=||fraction of the total formation volume that is sand|
|Vsh||=||fraction of the total formation volume that is shale|
|ρma||=||density of the rock matrix|
|σm||=||conductivity of the mud column, mS/m|
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SI Metric Conversion Factor
|cycles/sec||×||1.0*||E + 00||=||Hz|
|ft||×||3.048*||E − 01||=||m|
|ft2||×||9.290 304*||E − 02||=||m2|
|in.||×||2.54*||E + 00||=||cm|
|in.3||×||1.638 706||E + 01||=||cm3|
Conversion factor is exact.